Arrhenius Relationship

Introduction

The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

Introduction

The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

Life Stress Plots

The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (arrhenius) or:

[math]\displaystyle{ ln(L(V))=ln(C)+\frac{B}{V} }[/math]

[math]\displaystyle{ }[/math]

Arrhenius plot for Weibull life distribution.

In Eqn. (log-arrh) [math]\displaystyle{ \ln (C) }[/math] is the intercept of the line and [math]\displaystyle{ B }[/math] is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Fig. 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (log-arrh) was plotted on a reciprocal scale. On such a scale, the slope [math]\displaystyle{ B }[/math] appears to be negative even though it has a positive value. This is because [math]\displaystyle{ B }[/math] is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( [math]\displaystyle{ \tfrac{1}{V} }[/math] is decreasing as [math]\displaystyle{ V }[/math] is increasing). The two different axes are shown in Fig. 3.

[math]\displaystyle{ }[/math]

An illustration of both reciprocal and non-reciprocal scales.

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Fig. 3 it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life. The shaded areas shown in Fig. 3 are the imposed at each test stress level. From such imposed [math]\displaystyle{ pdfs }[/math] one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Fig. 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[math]\displaystyle{ }[/math]

An example of scatter in life at each test stress level.

Activation Energy and the Parameter B

Depending on the application (and where the stress is exclusively thermal), the parameter [math]\displaystyle{ B }[/math] can be replaced by:

[math]\displaystyle{ B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzma}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.623\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}} }[/math]

Note that in this formulation, the activation energy [math]\displaystyle{ {{E}_{A}} }[/math] must be known a priori. If the activation energy is known then there is only one model parameter remaining, [math]\displaystyle{ C. }[/math] Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat [math]\displaystyle{ B }[/math] as one of the model parameters. As it can be seen in Eqn. (arrhenius), [math]\displaystyle{ B }[/math] has the same properties as the activation energy. In other words, [math]\displaystyle{ B }[/math] is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of [math]\displaystyle{ B, }[/math] the higher the dependency of the life on the specific stress (see Fig. 5). Parameter [math]\displaystyle{ B }[/math] may also take negative values. In that case, life is increasing with increasing stress (see Fig. 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[math]\displaystyle{ }[/math]

Behavior of the parameter B.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed jointly and with a complete MLE solution over the entire data set, using ReliaSoft's ALTA. The analysis yields:

[math]\displaystyle{ \widehat{\beta }=4.2915822 }[/math]

[math]\displaystyle{ \widehat{B}=1861.6186657 }[/math]

[math]\displaystyle{ \widehat{C}=58.9848692 }[/math]

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Fig. 10 that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations for confidence intervals are presented in Appendix A. Specific calculations for confidence bounds on the Arrhenius model are presented in Appendix 6.A following this chapter.)

Comparison of the confidence bounds for different use stress levels.

Appendix 6.A: Arrhenius Confidence Bounds

Approximate Confidence Bounds for the Arrhenius-Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting [math]\displaystyle{ m=L(V) }[/math] as shown in Eqn. (Arrean). The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life are then estimated by:

[math]\displaystyle{ \begin{align} & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\ & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level (i.e., 95%=0.95), then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \begin{align} & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right] }[/math]

The variances and covariance of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B} }[/math] , [math]\displaystyle{ \widehat{C}) }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The bounds on reliability for any given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Confidence Bounds on Time

The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are then estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Approximate Confidence Bounds for the Arrhenius-Weibull:

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \begin{align} & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align} }[/math]

also:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}} }[/math]

or:

[math]\displaystyle{ \widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}} }[/math]

Setting:

[math]\displaystyle{ \widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right] }[/math]

or:

[math]\displaystyle{ \widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right] }[/math]

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ \widehat{u}\ \ : }[/math]

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\ & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Confidence Bounds on Time

The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \begin{align} & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T} }[/math] .

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on time can then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on [math]\displaystyle{ B }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and the parameter [math]\displaystyle{ C }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (C) }[/math] are treated as normally distributed. The bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]= }[/math]

[math]\displaystyle{ ={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is:

[math]\displaystyle{ R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Let [math]\displaystyle{ \widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}, }[/math] then .. For [math]\displaystyle{ t={T}' }[/math] , [math]\displaystyle{ \widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} }[/math] , and for [math]\displaystyle{ t=\infty , }[/math] [math]\displaystyle{ \widehat{z}=\infty . }[/math] The above equation then becomes:

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ \begin{align} & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align} }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The next step is to calculate the variance of [math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}): }[/math]

[math]\displaystyle{ \begin{align} & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

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Life Stress Plots

The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (arrhenius) or:

[math]\displaystyle{ ln(L(V))=ln(C)+\frac{B}{V} }[/math]

[math]\displaystyle{ }[/math]

Arrhenius plot for Weibull life distribution.

In Eqn. (log-arrh) [math]\displaystyle{ \ln (C) }[/math] is the intercept of the line and [math]\displaystyle{ B }[/math] is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Fig. 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (log-arrh) was plotted on a reciprocal scale. On such a scale, the slope [math]\displaystyle{ B }[/math] appears to be negative even though it has a positive value. This is because [math]\displaystyle{ B }[/math] is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( [math]\displaystyle{ \tfrac{1}{V} }[/math] is decreasing as [math]\displaystyle{ V }[/math] is increasing). The two different axes are shown in Fig. 3.

[math]\displaystyle{ }[/math]

An illustration of both reciprocal and non-reciprocal scales.

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Fig. 3 it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life. The shaded areas shown in Fig. 3 are the imposed at each test stress level. From such imposed [math]\displaystyle{ pdfs }[/math] one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Fig. 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[math]\displaystyle{ }[/math]

An example of scatter in life at each test stress level.

Activation Energy and the Parameter B

Depending on the application (and where the stress is exclusively thermal), the parameter [math]\displaystyle{ B }[/math] can be replaced by:

[math]\displaystyle{ B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzma}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.623\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}} }[/math]

Note that in this formulation, the activation energy [math]\displaystyle{ {{E}_{A}} }[/math] must be known a priori. If the activation energy is known then there is only one model parameter remaining, [math]\displaystyle{ C. }[/math] Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat [math]\displaystyle{ B }[/math] as one of the model parameters. As it can be seen in Eqn. (arrhenius), [math]\displaystyle{ B }[/math] has the same properties as the activation energy. In other words, [math]\displaystyle{ B }[/math] is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of [math]\displaystyle{ B, }[/math] the higher the dependency of the life on the specific stress (see Fig. 5). Parameter [math]\displaystyle{ B }[/math] may also take negative values. In that case, life is increasing with increasing stress (see Fig. 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[math]\displaystyle{ }[/math]

Behavior of the parameter B.

Introduction

The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

Life Stress Plots

The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (arrhenius) or:

[math]\displaystyle{ ln(L(V))=ln(C)+\frac{B}{V} }[/math]

[math]\displaystyle{ }[/math]

Arrhenius plot for Weibull life distribution.

In Eqn. (log-arrh) [math]\displaystyle{ \ln (C) }[/math] is the intercept of the line and [math]\displaystyle{ B }[/math] is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Fig. 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (log-arrh) was plotted on a reciprocal scale. On such a scale, the slope [math]\displaystyle{ B }[/math] appears to be negative even though it has a positive value. This is because [math]\displaystyle{ B }[/math] is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( [math]\displaystyle{ \tfrac{1}{V} }[/math] is decreasing as [math]\displaystyle{ V }[/math] is increasing). The two different axes are shown in Fig. 3.

[math]\displaystyle{ }[/math]

An illustration of both reciprocal and non-reciprocal scales.

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Fig. 3 it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life. The shaded areas shown in Fig. 3 are the imposed at each test stress level. From such imposed [math]\displaystyle{ pdfs }[/math] one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Fig. 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[math]\displaystyle{ }[/math]

An example of scatter in life at each test stress level.

Activation Energy and the Parameter B

Depending on the application (and where the stress is exclusively thermal), the parameter [math]\displaystyle{ B }[/math] can be replaced by:

[math]\displaystyle{ B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzma}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.623\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}} }[/math]

Note that in this formulation, the activation energy [math]\displaystyle{ {{E}_{A}} }[/math] must be known a priori. If the activation energy is known then there is only one model parameter remaining, [math]\displaystyle{ C. }[/math] Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat [math]\displaystyle{ B }[/math] as one of the model parameters. As it can be seen in Eqn. (arrhenius), [math]\displaystyle{ B }[/math] has the same properties as the activation energy. In other words, [math]\displaystyle{ B }[/math] is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of [math]\displaystyle{ B, }[/math] the higher the dependency of the life on the specific stress (see Fig. 5). Parameter [math]\displaystyle{ B }[/math] may also take negative values. In that case, life is increasing with increasing stress (see Fig. 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[math]\displaystyle{ }[/math]

Behavior of the parameter B.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed jointly and with a complete MLE solution over the entire data set, using ReliaSoft's ALTA. The analysis yields:

[math]\displaystyle{ \widehat{\beta }=4.2915822 }[/math]

[math]\displaystyle{ \widehat{B}=1861.6186657 }[/math]

[math]\displaystyle{ \widehat{C}=58.9848692 }[/math]

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Fig. 10 that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations for confidence intervals are presented in Appendix A. Specific calculations for confidence bounds on the Arrhenius model are presented in Appendix 6.A following this chapter.)

Comparison of the confidence bounds for different use stress levels.

Appendix 6.A: Arrhenius Confidence Bounds

Approximate Confidence Bounds for the Arrhenius-Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting [math]\displaystyle{ m=L(V) }[/math] as shown in Eqn. (Arrean). The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life are then estimated by:

[math]\displaystyle{ \begin{align} & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\ & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level (i.e., 95%=0.95), then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \begin{align} & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right] }[/math]

The variances and covariance of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B} }[/math] , [math]\displaystyle{ \widehat{C}) }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The bounds on reliability for any given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Confidence Bounds on Time

The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are then estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Approximate Confidence Bounds for the Arrhenius-Weibull:

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \begin{align} & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align} }[/math]

also:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}} }[/math]

or:

[math]\displaystyle{ \widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}} }[/math]

Setting:

[math]\displaystyle{ \widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right] }[/math]

or:

[math]\displaystyle{ \widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right] }[/math]

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ \widehat{u}\ \ : }[/math]

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\ & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Confidence Bounds on Time

The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \begin{align} & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T} }[/math] .

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on time can then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on [math]\displaystyle{ B }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and the parameter [math]\displaystyle{ C }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (C) }[/math] are treated as normally distributed. The bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]= }[/math]

[math]\displaystyle{ ={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is:

[math]\displaystyle{ R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Let [math]\displaystyle{ \widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}, }[/math] then .. For [math]\displaystyle{ t={T}' }[/math] , [math]\displaystyle{ \widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} }[/math] , and for [math]\displaystyle{ t=\infty , }[/math] [math]\displaystyle{ \widehat{z}=\infty . }[/math] The above equation then becomes:

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ \begin{align} & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align} }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The next step is to calculate the variance of [math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}): }[/math]

[math]\displaystyle{ \begin{align} & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

The material on this page is copyrighted.

©1992-2012. ReliaSoft Corporation. ALL RIGHTS RESERVED.

Introduction

The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

Life Stress Plots

The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (arrhenius) or:

[math]\displaystyle{ ln(L(V))=ln(C)+\frac{B}{V} }[/math]

[math]\displaystyle{ }[/math]

Arrhenius plot for Weibull life distribution.

In Eqn. (log-arrh) [math]\displaystyle{ \ln (C) }[/math] is the intercept of the line and [math]\displaystyle{ B }[/math] is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Fig. 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (log-arrh) was plotted on a reciprocal scale. On such a scale, the slope [math]\displaystyle{ B }[/math] appears to be negative even though it has a positive value. This is because [math]\displaystyle{ B }[/math] is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( [math]\displaystyle{ \tfrac{1}{V} }[/math] is decreasing as [math]\displaystyle{ V }[/math] is increasing). The two different axes are shown in Fig. 3.

[math]\displaystyle{ }[/math]

An illustration of both reciprocal and non-reciprocal scales.

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Fig. 3 it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life. The shaded areas shown in Fig. 3 are the imposed at each test stress level. From such imposed [math]\displaystyle{ pdfs }[/math] one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Fig. 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[math]\displaystyle{ }[/math]

An example of scatter in life at each test stress level.

Activation Energy and the Parameter B

Depending on the application (and where the stress is exclusively thermal), the parameter [math]\displaystyle{ B }[/math] can be replaced by:

[math]\displaystyle{ B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzma}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.623\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}} }[/math]

Note that in this formulation, the activation energy [math]\displaystyle{ {{E}_{A}} }[/math] must be known a priori. If the activation energy is known then there is only one model parameter remaining, [math]\displaystyle{ C. }[/math] Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat [math]\displaystyle{ B }[/math] as one of the model parameters. As it can be seen in Eqn. (arrhenius), [math]\displaystyle{ B }[/math] has the same properties as the activation energy. In other words, [math]\displaystyle{ B }[/math] is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of [math]\displaystyle{ B, }[/math] the higher the dependency of the life on the specific stress (see Fig. 5). Parameter [math]\displaystyle{ B }[/math] may also take negative values. In that case, life is increasing with increasing stress (see Fig. 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[math]\displaystyle{ }[/math]

Behavior of the parameter B.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed jointly and with a complete MLE solution over the entire data set, using ReliaSoft's ALTA. The analysis yields:

[math]\displaystyle{ \widehat{\beta }=4.2915822 }[/math]

[math]\displaystyle{ \widehat{B}=1861.6186657 }[/math]

[math]\displaystyle{ \widehat{C}=58.9848692 }[/math]

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Fig. 10 that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations for confidence intervals are presented in Appendix A. Specific calculations for confidence bounds on the Arrhenius model are presented in Appendix 6.A following this chapter.)

Comparison of the confidence bounds for different use stress levels.

Appendix 6.A: Arrhenius Confidence Bounds

Approximate Confidence Bounds for the Arrhenius-Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting [math]\displaystyle{ m=L(V) }[/math] as shown in Eqn. (Arrean). The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life are then estimated by:

[math]\displaystyle{ \begin{align} & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\ & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level (i.e., 95%=0.95), then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \begin{align} & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right] }[/math]

The variances and covariance of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B} }[/math] , [math]\displaystyle{ \widehat{C}) }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The bounds on reliability for any given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Confidence Bounds on Time

The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are then estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Approximate Confidence Bounds for the Arrhenius-Weibull:

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \begin{align} & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align} }[/math]

also:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}} }[/math]

or:

[math]\displaystyle{ \widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}} }[/math]

Setting:

[math]\displaystyle{ \widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right] }[/math]

or:

[math]\displaystyle{ \widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right] }[/math]

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ \widehat{u}\ \ : }[/math]

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\ & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Confidence Bounds on Time

The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \begin{align} & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T} }[/math] .

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on time can then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on [math]\displaystyle{ B }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and the parameter [math]\displaystyle{ C }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (C) }[/math] are treated as normally distributed. The bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]= }[/math]

[math]\displaystyle{ ={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is:

[math]\displaystyle{ R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Let [math]\displaystyle{ \widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}, }[/math] then .. For [math]\displaystyle{ t={T}' }[/math] , [math]\displaystyle{ \widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} }[/math] , and for [math]\displaystyle{ t=\infty , }[/math] [math]\displaystyle{ \widehat{z}=\infty . }[/math] The above equation then becomes:

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ \begin{align} & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align} }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The next step is to calculate the variance of [math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}): }[/math]

[math]\displaystyle{ \begin{align} & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

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©1992-2012. ReliaSoft Corporation. ALL RIGHTS RESERVED.

Introduction

The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

Life Stress Plots

The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (arrhenius) or:

[math]\displaystyle{ ln(L(V))=ln(C)+\frac{B}{V} }[/math]

[math]\displaystyle{ }[/math]

Arrhenius plot for Weibull life distribution.

In Eqn. (log-arrh) [math]\displaystyle{ \ln (C) }[/math] is the intercept of the line and [math]\displaystyle{ B }[/math] is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Fig. 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (log-arrh) was plotted on a reciprocal scale. On such a scale, the slope [math]\displaystyle{ B }[/math] appears to be negative even though it has a positive value. This is because [math]\displaystyle{ B }[/math] is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( [math]\displaystyle{ \tfrac{1}{V} }[/math] is decreasing as [math]\displaystyle{ V }[/math] is increasing). The two different axes are shown in Fig. 3.

[math]\displaystyle{ }[/math]

An illustration of both reciprocal and non-reciprocal scales.

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Fig. 3 it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life. The shaded areas shown in Fig. 3 are the imposed at each test stress level. From such imposed [math]\displaystyle{ pdfs }[/math] one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Fig. 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[math]\displaystyle{ }[/math]

An example of scatter in life at each test stress level.

Activation Energy and the Parameter B

Depending on the application (and where the stress is exclusively thermal), the parameter [math]\displaystyle{ B }[/math] can be replaced by:

[math]\displaystyle{ B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzma}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.623\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}} }[/math]

Note that in this formulation, the activation energy [math]\displaystyle{ {{E}_{A}} }[/math] must be known a priori. If the activation energy is known then there is only one model parameter remaining, [math]\displaystyle{ C. }[/math] Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat [math]\displaystyle{ B }[/math] as one of the model parameters. As it can be seen in Eqn. (arrhenius), [math]\displaystyle{ B }[/math] has the same properties as the activation energy. In other words, [math]\displaystyle{ B }[/math] is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of [math]\displaystyle{ B, }[/math] the higher the dependency of the life on the specific stress (see Fig. 5). Parameter [math]\displaystyle{ B }[/math] may also take negative values. In that case, life is increasing with increasing stress (see Fig. 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[math]\displaystyle{ }[/math]

Behavior of the parameter B.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed jointly and with a complete MLE solution over the entire data set, using ReliaSoft's ALTA. The analysis yields:

[math]\displaystyle{ \widehat{\beta }=4.2915822 }[/math]

[math]\displaystyle{ \widehat{B}=1861.6186657 }[/math]

[math]\displaystyle{ \widehat{C}=58.9848692 }[/math]

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Fig. 10 that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations for confidence intervals are presented in Appendix A. Specific calculations for confidence bounds on the Arrhenius model are presented in Appendix 6.A following this chapter.)

Comparison of the confidence bounds for different use stress levels.

Appendix 6.A: Arrhenius Confidence Bounds

Approximate Confidence Bounds for the Arrhenius-Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting [math]\displaystyle{ m=L(V) }[/math] as shown in Eqn. (Arrean). The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life are then estimated by:

[math]\displaystyle{ \begin{align} & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\ & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level (i.e., 95%=0.95), then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \begin{align} & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right] }[/math]

The variances and covariance of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B} }[/math] , [math]\displaystyle{ \widehat{C}) }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The bounds on reliability for any given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Confidence Bounds on Time

The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are then estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Approximate Confidence Bounds for the Arrhenius-Weibull:

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \begin{align} & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align} }[/math]

also:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}} }[/math]

or:

[math]\displaystyle{ \widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}} }[/math]

Setting:

[math]\displaystyle{ \widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right] }[/math]

or:

[math]\displaystyle{ \widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right] }[/math]

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ \widehat{u}\ \ : }[/math]

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\ & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Confidence Bounds on Time

The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \begin{align} & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T} }[/math] .

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on time can then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on [math]\displaystyle{ B }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and the parameter [math]\displaystyle{ C }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (C) }[/math] are treated as normally distributed. The bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]= }[/math]

[math]\displaystyle{ ={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is:

[math]\displaystyle{ R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Let [math]\displaystyle{ \widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}, }[/math] then .. For [math]\displaystyle{ t={T}' }[/math] , [math]\displaystyle{ \widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} }[/math] , and for [math]\displaystyle{ t=\infty , }[/math] [math]\displaystyle{ \widehat{z}=\infty . }[/math] The above equation then becomes:

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ \begin{align} & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align} }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The next step is to calculate the variance of [math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}): }[/math]

[math]\displaystyle{ \begin{align} & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

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©1992-2012. ReliaSoft Corporation. ALL RIGHTS RESERVED.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed jointly and with a complete MLE solution over the entire data set, using ReliaSoft's ALTA. The analysis yields:

[math]\displaystyle{ \widehat{\beta }=4.2915822 }[/math]

[math]\displaystyle{ \widehat{B}=1861.6186657 }[/math]

[math]\displaystyle{ \widehat{C}=58.9848692 }[/math]

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Fig. 10 that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations for confidence intervals are presented in Appendix A. Specific calculations for confidence bounds on the Arrhenius model are presented in Appendix 6.A following this chapter.)

Comparison of the confidence bounds for different use stress levels.

Introduction

The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

Life Stress Plots

The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (arrhenius) or:

[math]\displaystyle{ ln(L(V))=ln(C)+\frac{B}{V} }[/math]

[math]\displaystyle{ }[/math]

Arrhenius plot for Weibull life distribution.

In Eqn. (log-arrh) [math]\displaystyle{ \ln (C) }[/math] is the intercept of the line and [math]\displaystyle{ B }[/math] is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Fig. 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (log-arrh) was plotted on a reciprocal scale. On such a scale, the slope [math]\displaystyle{ B }[/math] appears to be negative even though it has a positive value. This is because [math]\displaystyle{ B }[/math] is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( [math]\displaystyle{ \tfrac{1}{V} }[/math] is decreasing as [math]\displaystyle{ V }[/math] is increasing). The two different axes are shown in Fig. 3.

[math]\displaystyle{ }[/math]

An illustration of both reciprocal and non-reciprocal scales.

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Fig. 3 it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life. The shaded areas shown in Fig. 3 are the imposed at each test stress level. From such imposed [math]\displaystyle{ pdfs }[/math] one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Fig. 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[math]\displaystyle{ }[/math]

An example of scatter in life at each test stress level.

Activation Energy and the Parameter B

Depending on the application (and where the stress is exclusively thermal), the parameter [math]\displaystyle{ B }[/math] can be replaced by:

[math]\displaystyle{ B=\frac{{{E}_{A}}}{K}=\frac{\text{activation energy}}{\text{Boltzma}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.623\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}} }[/math]

Note that in this formulation, the activation energy [math]\displaystyle{ {{E}_{A}} }[/math] must be known a priori. If the activation energy is known then there is only one model parameter remaining, [math]\displaystyle{ C. }[/math] Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat [math]\displaystyle{ B }[/math] as one of the model parameters. As it can be seen in Eqn. (arrhenius), [math]\displaystyle{ B }[/math] has the same properties as the activation energy. In other words, [math]\displaystyle{ B }[/math] is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of [math]\displaystyle{ B, }[/math] the higher the dependency of the life on the specific stress (see Fig. 5). Parameter [math]\displaystyle{ B }[/math] may also take negative values. In that case, life is increasing with increasing stress (see Fig. 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[math]\displaystyle{ }[/math]

Behavior of the parameter B.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed jointly and with a complete MLE solution over the entire data set, using ReliaSoft's ALTA. The analysis yields:

[math]\displaystyle{ \widehat{\beta }=4.2915822 }[/math]

[math]\displaystyle{ \widehat{B}=1861.6186657 }[/math]

[math]\displaystyle{ \widehat{C}=58.9848692 }[/math]

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Fig. 10 that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations for confidence intervals are presented in Appendix A. Specific calculations for confidence bounds on the Arrhenius model are presented in Appendix 6.A following this chapter.)

Comparison of the confidence bounds for different use stress levels.

Appendix 6.A: Arrhenius Confidence Bounds

Approximate Confidence Bounds for the Arrhenius-Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting [math]\displaystyle{ m=L(V) }[/math] as shown in Eqn. (Arrean). The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life are then estimated by:

[math]\displaystyle{ \begin{align} & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\ & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level (i.e., 95%=0.95), then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \begin{align} & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right] }[/math]

The variances and covariance of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B} }[/math] , [math]\displaystyle{ \widehat{C}) }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The bounds on reliability for any given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Confidence Bounds on Time

The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are then estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Approximate Confidence Bounds for the Arrhenius-Weibull:

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \begin{align} & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align} }[/math]

also:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}} }[/math]

or:

[math]\displaystyle{ \widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}} }[/math]

Setting:

[math]\displaystyle{ \widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right] }[/math]

or:

[math]\displaystyle{ \widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right] }[/math]

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ \widehat{u}\ \ : }[/math]

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\ & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Confidence Bounds on Time

The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \begin{align} & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T} }[/math] .

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on time can then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on [math]\displaystyle{ B }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and the parameter [math]\displaystyle{ C }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (C) }[/math] are treated as normally distributed. The bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]= }[/math]

[math]\displaystyle{ ={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is:

[math]\displaystyle{ R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Let [math]\displaystyle{ \widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}, }[/math] then .. For [math]\displaystyle{ t={T}' }[/math] , [math]\displaystyle{ \widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} }[/math] , and for [math]\displaystyle{ t=\infty , }[/math] [math]\displaystyle{ \widehat{z}=\infty . }[/math] The above equation then becomes:

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ \begin{align} & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align} }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The next step is to calculate the variance of [math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}): }[/math]

[math]\displaystyle{ \begin{align} & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

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Appendix 6.A: Arrhenius Confidence Bounds

Approximate Confidence Bounds for the Arrhenius-Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting [math]\displaystyle{ m=L(V) }[/math] as shown in Eqn. (Arrean). The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life are then estimated by:

[math]\displaystyle{ \begin{align} & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\ & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level (i.e., 95%=0.95), then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \begin{align} & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right] }[/math]

The variances and covariance of [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B} }[/math] , [math]\displaystyle{ \widehat{C}) }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The bounds on reliability for any given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Confidence Bounds on Time

The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are then estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

where [math]\displaystyle{ {{m}_{U}} }[/math] and [math]\displaystyle{ {{m}_{L}} }[/math] are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).

Approximate Confidence Bounds for the Arrhenius-Weibull:

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \begin{align} & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align} }[/math]

also:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Confidence Bounds on Reliability

The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}} }[/math]

or:

[math]\displaystyle{ \widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}} }[/math]

Setting:

[math]\displaystyle{ \widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right] }[/math]

or:

[math]\displaystyle{ \widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right] }[/math]

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ \widehat{u}\ \ : }[/math]

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\ & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Confidence Bounds on Time

The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

[math]\displaystyle{ \begin{align} & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T} }[/math] .

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align} }[/math]

The upper and lower bounds on time can then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

where [math]\displaystyle{ {{u}_{U}} }[/math] and [math]\displaystyle{ {{u}_{L}} }[/math] are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on [math]\displaystyle{ B }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and the parameter [math]\displaystyle{ C }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (C) }[/math] are treated as normally distributed. The bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} \end{align} }[/math]

The variances and covariances of [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left[ \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]= }[/math]

[math]\displaystyle{ ={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is:

[math]\displaystyle{ R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

Let [math]\displaystyle{ \widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}, }[/math] then .. For [math]\displaystyle{ t={T}' }[/math] , [math]\displaystyle{ \widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} }[/math] , and for [math]\displaystyle{ t=\infty , }[/math] [math]\displaystyle{ \widehat{z}=\infty . }[/math] The above equation then becomes:

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ \begin{align} & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align} }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The next step is to calculate the variance of [math]\displaystyle{ {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}): }[/math]

[math]\displaystyle{ \begin{align} & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

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