Eyring Relationship

Eyring Relationship

Introduction

The Eyring relationship was formulated from quantum mechanics principles [9] and is most often used when thermal stress (temperature) is the acceleration variable. However, the Eyring relationship is also often used for stress variables other than temperature, such as humidity. The relationship is given by:

$L(V)=\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

where:

• $L$ represents a quantifiable life measure, such as mean life, characteristic life, median life, $B(x)$ life, etc.

• $V$ represents the stress level (temperature values in absolute units, i.e. degrees Kelvin or degrees Rankine ).

• $A$ is one of the model parameters to be determined.

• $B$ is another model parameter to be determined.

Graphical look at the Eyring relationship (linear scale), at different life characteristics and with a Weibull life distribution.

The Eyring relationship is similar to the Arrhenius relationship. This similarity is more apparent if Eqn. (eyring) is rewritten in the following way:

$\begin{array}{rlr}& L(V)=& \frac{1}{V}{e}^{-(A-\frac{B}{V})}\\ & =& \frac{e^{-A}}{V}{e}^{\frac{B}{V}}\end{array}$

or:

$L(V)=\frac{1}{V}Const.\cdot e^{\frac{B}{V}}$

The Arrhenius relationship is given by:

$L(V)=C\cdot e^{\frac{B}{V}}$

Comparing Eqn. (eyr-arr) to the Arrhenius relationship, it can be seen that the only difference between the two relationships is the $\frac{1}{V}$ term in Eqn. (eyr-arr). In general, both relationships yield very similar results. Like the Arrhenius, the Eyring relationship is plotted on a log-reciprocal paper.

Eyring relationship plotted on Arrhenius paper.

Acceleration Factor

For the Eyring model the acceleration factor is given by:

$A_F=\frac{L_{USE}}{L_{Accelerated}}=\frac{\frac{1}{V_u}{e}^{-(A-\frac{B}{V_u})}}{\frac{1}{V_A}{e}^{-(A-\frac{B}{V_A})}}=\frac{e^{\frac{B}{V_u}}}{e^{\frac{B}{V_A}}}=\frac{V_A}{V_u}{e}^{B(\frac{1}{V_u}-\frac{1}{V_A})}$

Eyring-Exponential

The $pdf$ of the 1-parameter exponential distribution is given by:

$f(t)=\lambda \cdot e^{-\lambda \cdot t}$

It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:

$\lambda =\frac{1}{m}$

thus:

$f(t)=\frac{1}{m}\cdot e^{-\frac{t}{m}}$

The Eyring-exponential model $pdf$ can then be obtained by setting $m=L(V)$ in Eqn. (eyring):

$m=L(V)=\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

and substituting for $m$ in Eqn. (pdfexpm2):

$f(t,V)=V\cdot e^{(A-\frac{B}{V})}{e}^{-V\cdot e^{(A-\frac{B}{V})}\cdot t}$

Eyring-Exponential Statistical Properties Summary

Mean or MTTF

The mean, $\bar{T},$ or Mean Time To Failure (MTTF) for the Eyring-exponential is given by:

$\begin{array}{rlr}& \bar{T}=& \underset{0}{\overset{\mathrm{\infty }}{}}t\cdot f(t,V)dt=\underset{0}{\overset{\mathrm{\infty }}{}}t\cdot V{e}^{(A-\frac{B}{V})}{e}^{-tV{e}^{(A-\frac{B}{V})}}dt\\ & =& \frac{1}{V}{e}^{-(A-\frac{B}{V})}\end{array}$

Median

The median, $\breve{T},$ for the Eyring-exponential model is given by:

$\breve{T}=0.693\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

Mode

The mode, $\tilde{T},$ for the Eyring-exponential model is $\tilde{T}=0.$

Standard Deviation

The standard deviation, ${\sigma }_T$, for the Eyring-exponential model is given by:

${\sigma }_T=\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

Eyring-Exponential Reliability Function

The Eyring-exponential reliability function is given by:

$R(T,V)=e^{-T\cdot V\cdot e^{(A-\frac{B}{V})}}$

This function is the complement of the Eyring-exponential cumulative distribution function or:

$R(T,V)=1-Q(T,V)=1-\underset{0}{\overset{T}{}}f(T,V)dT$

and:

$R(T,V)=1-\underset{0}{\overset{T}{}}V{e}^{(A-\frac{B}{V})}{e}^{-T\cdot V\cdot e^{(A-\frac{B}{V})}}dT=e^{-T\cdot V\cdot e^{(A-\frac{B}{V})}}$

Conditional Reliability

The conditional reliability function for the Eyring-exponential model is given by:

$R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{e^{-\lambda (T+t)}}{e^{-\lambda T}}=e^{-t\cdot V\cdot e^{(A-\frac{B}{V})}}$

Reliable Life

For the Eyring-exponential model, the reliable life, or the mission duration for a desired reliability goal, $t_{R,}$ is given by:

$R(t_R,V)=e^{-t_R\cdot V\cdot e^{(A-\frac{B}{V})}}$

$\ln [R(t_R,V)]=-t_R\cdot V\cdot e^{(A-\frac{B}{V})}$

or:

$t_R=-\frac{1}{V}{e}^{-(A-\frac{B}{V})}\ln [R(t_R,V)]$

Parameter Estimation

Maximum Likelihood Estimation Method

The complete exponential log-likelihood function of the Eyring model is composed of two summation portions:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln [V_i\cdot e^{(A-\frac{B}{V_i})}{e}^{-V_i\cdot e^{(A-\frac{B}{V_i})}\cdot T_i}]\\ & & -\underset{i=1}{\sum ^S}{N}_i^{\prime }\cdot V_i\cdot e^{(A-\frac{B}{V_i})}\cdot T_i^{\prime }+\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]\end{array}$

where:

$R_{Li}^{\prime \prime }=e^{-T_{Li}^{\prime \prime }{V}_i{e}^{A-\frac{B}{V_i}}}$

$R_{Ri}^{\prime \prime }=e^{-T_{Ri}^{\prime \prime }{V}_i{e}^{A-\frac{B}{V_i}}}$

and:

• $F_e$ is the number of groups of exact times-to-failure data points.
• $N_i$ is the number of times-to-failure in the $i^{th}$ time-to-failure data group.
• $V_i$ is the stress level of the $i^{th}$ group.
• $A$ is the Eyring parameter (unknown, the first of two parameters to be estimated).
• $B$ is the second Eyring parameter (unknown, the second of two parameters to be estimated).
• $T_i$ is the exact failure time of the $i^{th}$ group.
• $S$ is the number of groups of suspension data points.
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points.
• $T_i^{\prime }$ is the running time of the $i^{th}$ suspension data group.
• $FI$ is the number of interval data groups.
• $N_i^{\prime \prime }$ is the number of intervals in the i ${}^{th}$ group of data intervals.
• $T_{Li}^{\prime \prime }$ is the beginning of the i ${}^{th}$ interval.
• $T_{Ri}^{\prime \prime }$ is the ending of the i ${}^{th}$ interval.
The solution (parameter estimates) will be found by solving for the parameters $\hat{A}$ and $\hat{B}$ so that $\frac{\partial \mathrm{\Lambda }}{\partial A}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial B}=0$ where:

$\begin{array}{rlr}& \frac{\partial \mathrm{\Lambda }}{\partial A}=& \underset{i=1}{\sum ^{F_e}}{N}_i(1-V_i\cdot e^{(A-\frac{B}{V_i})}{T}_i)-\underset{i=1}{\sum ^S}{N}_i^{\prime }{V}_i\cdot e^{(A-\frac{B}{V_i})}{T}_i^{\prime }\\ & & \stackrel{FI}{\underset{i=1}{-\sum ^{}_{}}}{N}_i^{\prime \prime }\frac{(T_{Li}^{\prime \prime }{R}_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }{R}_{Ri}^{\prime \prime })V_i{e}^{A-\frac{B}{V_i}}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}\end{array}$

$\begin{array}{rlr}& \frac{\partial \mathrm{\Lambda }}{\partial B}=& \underset{i=1}{\sum ^{F_e}}{N}_i[e^{(A-\frac{B}{V_i})}{T}_i-\frac{1}{V_i}]+\underset{i=1}{\sum ^S}{N}_i^{\prime }\cdot e^{(A-\frac{B}{V_i})}{T}_i^{\prime }\\ & & \stackrel{FI}{\underset{i=1}{+\sum ^{}_{}}}{N}_i^{\prime \prime }\frac{(T_{Li}^{\prime \prime }{R}_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }{R}_{Ri}^{\prime \prime })e^{A-\frac{B}{V_i}}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}\end{array}$

Eyring-Weibull

The $pdf$ for 2-parameter Weibull distribution is given by:

$f(t)=\frac{\beta }{\eta }{\left(\frac{t}{\eta }\right)}^{\beta -1}{e}^{-{\left(\frac{t}{\eta }\right)}^{\beta }}$

The scale parameter (or characteristic life) of the Weibull distribution is $\eta$ . The Eyring-Weibull model $pdf$ can then be obtained by setting $\eta =L(V)$ in Eqn. (eyring):

$\eta =L(V)=\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

or:

$\frac{1}{\eta }=V\cdot e^{(A-\frac{B}{V})}$

Substituting for $\eta$ into Eqn. (Eyrpdf):

$f(t,V)=\beta \cdot V\cdot e^{(A-\frac{B}{V})}{(t\cdot V\cdot e^{(A-\frac{B}{V})})}^{\beta -1}{e}^{-{(t\cdot V\cdot e^{(A-\frac{B}{V})})}^{\beta }}$

Eyring-Weibull Statistical Properties Summary

Mean or MTTF

The mean, $\bar{T}$, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is given by:

$\bar{T}=\frac{1}{V}{e}^{-(A-\frac{B}{V})}\cdot \mathrm{\Gamma }(\frac{1}{\beta }+1)$

where $\mathrm{\Gamma }(\frac{1}{\beta }+1)$ is the gamma function evaluated at the value of $(\frac{1}{\beta }+1)$ .

Median

The median, $\breve{T}$ for the Eyring-Weibull model is given by:

$\breve{T}=\frac{1}{V}{e}^{-(A-\frac{B}{V})}{(\ln 2)}^{\frac{1}{\beta }}$

Mode

The mode, $\tilde{T},$ for the Eyring-Weibull model is given by:

$\tilde{T}=\frac{1}{V}{e}^{-(A-\frac{B}{V})}{(1-\frac{1}{\beta })}^{\frac{1}{\beta }}$

Standard Deviation

The standard deviation, ${\sigma }_T,$ for the Eyring-Weibull model is given by:

${\sigma }_T=\frac{1}{V}{e}^{-(A-\frac{B}{V})}\cdot \sqrt{\mathrm{\Gamma }(\frac{2}{\beta }+1)-{\left(\mathrm{\Gamma }(\frac{1}{\beta }+1)\right)}^2}$

Eyring-Weibull Reliability Function

The Eyring-Weibull reliability function is given by:

$R(T,V)=e^{-{(V\cdot T\cdot e^{(A-\frac{B}{V})})}^{\beta }}$

Conditional Reliability Function

The Eyring-Weibull conditional reliability function at a specified stress level is given by:

$R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{e^{-{((T+t)\cdot V\cdot e^{(A-\frac{B}{V})})}^{\beta }}}{e^{-{(V\cdot T\cdot e^{(A-\frac{B}{V})})}^{\beta }}}$

or:

$R(T,t,V)=e^{-[{((T+t)\cdot V\cdot e^{(A-\frac{B}{V})})}^{\beta }-{(V\cdot T\cdot e^{(A-\frac{B}{V})})}^{\beta }]}$

Reliable Life

For the Eyring-Weibull model, the reliable life, $t_R$ , of a unit for a specified reliability and starting the mission at age zero is given by:

$t_R=\frac{1}{V}{e}^{-(A-\frac{B}{V})}{\{-\ln \left[R(T_R,V)\right]\}}^{\frac{1}{\beta }}$

Eyring-Weibull Failure Rate Function

The Eyring-Weibull failure rate function, $\lambda (T)$ , is given by:

$\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\beta {(T\cdot V\cdot e^{(A-\frac{B}{V})})}^{\beta -1}$

Parameter Estimation

Maximum Likelihood Estimation Method

The Eyring-Weibull log-likelihood function is composed of two summation portions:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln [\beta \cdot V_i\cdot e^{A-\frac{B}{V_i}}{\left(T_i{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta -1}{e}^{-{\left(T_i{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }}]\\ & & -\underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(V_i{e}^{A-\frac{B}{V_i}}{T}_i^{\prime }\right)}^{\beta }+\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]\end{array}$

where:

$R_{Li}^{\prime \prime }=e^{-{\left(T_{Li}^{\prime \prime }{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }}$

$R_{Ri}^{\prime \prime }=e^{-{\left(T_{Ri}^{\prime \prime }{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }}$

and:

• $F_e$ is the number of groups of exact times-to-failure data points.
• $N_i$ is the number of times-to-failure data points in the $i^{th}$ time-to-failure data group.
• $\beta$ is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
• $A$ is the Eyring parameter (unknown, the second of three parameters to be estimated).
• $B$ is the second Eyring parameter (unknown, the third of three parameters to be estimated).
• $V_i$ is the stress level of the $i^{th}$ group.
• $T_i$ is the exact failure time of the $i^{th}$ group.
• $S$ is the number of groups of suspension data points.
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points.
• $T_i^{\prime }$ is the running time of the $i^{th}$ suspension data group.
• $FI$ is the number of interval data groups.
• $N_i^{\prime \prime }$ is the number of intervals in the i ${}^{th}$ group of data intervals.
• $T_{Li}^{\prime \prime }$ is the beginning of the i ${}^{th}$ interval.
• $T_{Ri}^{\prime \prime }$ is the ending of the i ${}^{th}$ interval.

The solution (parameter estimates) will be found by solving for the parameters $\beta ,$ $A$ and $B$ so that $\frac{\partial \mathrm{\Lambda }}{\partial \beta }=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial A}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial B}=0$

where:

$\begin{array}{rlr}& \frac{\partial \mathrm{\Lambda }}{\partial A}=& \beta \underset{i=1}{\sum ^{F_e}}{N}_i-\beta \underset{i=1}{\sum ^{F_e}}{N}_i{\left(T_i{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }\\ & & -\beta \underset{i=1}{\sum ^S}{N}_i^{\prime }{\left(T_i^{\prime }{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }\\ & & \stackrel{FI}{-\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\frac{\beta V_i^{\beta }{e}^{A\beta -\frac{B\beta }{V_i}}[{(T_{Li}^{\prime \prime })}^{\beta }{R}_{Li}^{\prime \prime }-{(T_{Ri}^{\prime \prime })}^{\beta }{R}_{Ri}^{\prime \prime }]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}\end{array}$

$\begin{array}{rlr}& \frac{\partial \mathrm{\Lambda }}{\partial B}=& -\beta \underset{i=1}{\sum ^{F_e}}{N}_i\frac{1}{V_i}+\beta \underset{i=1}{\sum ^{F_e}}{N}_i\frac{1}{V_i}{\left(T_i{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }\\ & & +\beta \underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{1}{V_i}{\left(T_i^{\prime }{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }\\ & & +\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\frac{\beta V_i^{(\beta -1)}{e}^{A\beta -\frac{B\beta }{V_i}}[{(T_{Li}^{\prime \prime })}^{\beta }{R}_{Li}^{\prime \prime }-{(T_{Ri}^{\prime \prime })}^{\beta }{R}_{Ri}^{\prime \prime }]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}\end{array}$

$\frac{\partial \mathrm{\Lambda }}{\partial \beta }=\frac{1}{\beta }\sum _{i=1}^{F_e}{N}_i\frac{1}{V_i}+\sum _{i=1}^{F_e}{N}_{i}ln\left(T_i{V}_i{e}^{A-\frac{B}{V_i}}\right)$

$-\sum _{i=1}^{F_e}{N}_i{\left(T_i{V}_i{e}^{A-\frac{B}{V_i}}\right)}^{\beta }ln(T_{i}V)i{e}^{A-\frac{B}{V_i}})$

$\text{Missing open brace for superscript}$

$\text{Missing open brace for superscript}$

Example

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

Pdf of the lognormal distribution with different log-std values.

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

$\hat{\beta }=4.29186497$

$$

$\hat{B}=1454.08635742$

Once the parameters of the model are defined, other life measures can be directly obtained using the appropriate equations. For example, the MTTF can be obtained for the use stress level of 323K using:

$\bar{T}=\frac{1}{V}{e}^{-(A-\frac{B}{V})}\cdot \mathrm{\Gamma }(\frac{1}{\beta }+1)$

or:

$\begin{array}{rlr}& \bar{T}=& \frac{1}{323}{e}^{-(-11.08784624-\frac{1454.08635742}{323})}\cdot \mathrm{\Gamma }(\frac{1}{4.29186497}+1)\\ & =& 16,610hr\end{array}$

Eyring-Lognormal

The $pdf$ of the lognormal distribution is given by:

$f(T)=\frac{1}{T{\sigma }_{T^{\prime }}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{T^{\prime }-\overline{T^{\prime }}}{{\sigma }_{T^{\prime }}}\right)}^2}$

where:

$T^{\prime }=\ln (T)$

$T=\text{times-to-failure}$

and:

• $\overline{T^{\prime }}=$ mean of the natural logarithms of the times-to-failure.

• ${\sigma }_{T^{\prime }}=$ standard deviation of the natural logarithms of the times-to-failure.

The Eyring-lognormal model can be obtained first by setting $\breve{T}=L(V)$ in Eqn. (eyring). Therefore:

$\breve{T}=L(V)=\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

or:

$e^{{\bar{T}}^{\prime }}=\frac{1}{V}{e}^{-(A-\frac{B}{V})}$

Thus:

${\bar{T}}^{\prime }=-\ln (V)-A+\frac{B}{V}$

Substituting Eqn. (eyr-logn-mean) into Eqn. (Eyr-logn-pdf) yields the Eyring-lognormal model $pdf$

or:

$f(T,V)=\frac{1}{T{\sigma }_{T^{\prime }}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{T^{\prime }+\ln (V)+A-\frac{B}{V}}{{\sigma }_{T^{\prime }}}\right)}^2}$

Eyring-Lognormal Statistical Properties Summary

The Mean

• The mean life of the Eyring-lognormal model (mean of the times-to-failure), $\overline{T}$ , is given by:

$\begin{array}{rlr}& \overline{T}=& e^{\overline{T^{\prime }}+\frac{1}{2}{\sigma }_{T^{\prime }}^2}\\ & =& e^{-\ln (V)-A+\frac{B}{V}+\frac{1}{2}{\sigma }_{T^{\prime }}^2}\end{array}$

The mean of the natural logarithms of the times-to-failure, ${\overline{T}}^{{}^{\prime }}$ , in terms of $\overline{T}$ and ${\sigma }_T$ is given by:

${\overline{T}}^{\prime }=\ln \left(\overline{T}\right)-\frac{1}{2}\ln (\frac{{\sigma }_T^2}{{\overline{T}}^2}+1)$

The Median

The median of the Eyring-lognormal model is given by:

$\breve{T}=e^{{\bar{T}}^{\prime }}$

The Standard Deviation

• The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure), ${\sigma }_T$ , is given by:

$\begin{array}{rlr}& {\sigma }_T=& \sqrt{\left(e^{2\overline{T^{\prime }}+{\sigma }_{T^{\prime }}^2}\right)(e^{{\sigma }_{T^{\prime }}^2}-1)}\\ & =& \sqrt{\left(e^{2(-\ln (V)-A+\frac{B}{V})+{\sigma }_{T^{\prime }}^2}\right)(e^{{\sigma }_{T^{\prime }}^2}-1)}\end{array}$

• The standard deviation of the natural logarithms of the times-to-failure, ${\sigma }_{T^{\prime }}$ , in terms of $\overline{T}$ and ${\sigma }_T$ is given by:

${\sigma }_{T^{\prime }}=\sqrt{\ln (\frac{{\sigma }_T^2}{{\overline{T}}^2}+1)}$

The Mode

• The mode of the Eyring-lognormal model is given by:

$\begin{array}{rlr}& \tilde{T}=& e^{{\bar{T}}^{\prime }-{\sigma }_{T^{\prime }}^2}\\ & =& e^{-\ln (V)-A+\frac{B}{V}-{\sigma }_{T^{\prime }}^2}\end{array}$

Eyring-Lognormal Reliability Function

The reliability for a mission of time $T$ , starting at age 0, for the Eyring-lognormal model is determined by:

$R(T,V)=\underset{T}{\overset{\mathrm{\infty }}{}}f(t,V)dt$

or:

$R(T,V)=\underset{T^{{}^{\prime }}}{\overset{\mathrm{\infty }}{}}\frac{1}{{\sigma }_{T^{\prime }}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{t+\ln (V)+A-\frac{B}{V}}{{\sigma }_{T^{\prime }}}\right)}^2}dt$

There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods.

Reliable Life

For the Eyring-lognormal model, the reliable life, or the mission duration for a desired reliability goal, $t_R,$ is estimated by first solving the reliability equation with respect to time, as follows:

$T_R^{\prime }=-\ln (V)-A+\frac{B}{V}+z\cdot {\sigma }_{T^{\prime }}$

where:

$z={\mathrm{\Phi }}^{-1}\left[F(T_R^{\prime },V)\right]$

and:

$\mathrm{\Phi }(z)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{z(T^{\prime },V)}{}}{e}^{-\frac{t^2}{2}}dt$

Since $T^{\prime }=\ln (T)$ the reliable life, $t_{R,}$ is given by:

$t_R=e^{T_R^{\prime }}$

Eyring-Lognormal Failure Rate

The Eyring-lognormal failure rate is given by:

$\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\frac{1}{T{\sigma }_{T^{\prime }}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{T^{\prime }+\ln (V)+A-\frac{B}{V}}{{\sigma }_{T^{\prime }}}\right)}^2}}{{}_{T^{\prime }}^{\mathrm{\infty }}\frac{1}{{\sigma }_{T^{\prime }}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{T^{\prime }+\ln (V)+A-\frac{B}{V}}{{\sigma }_{T^{\prime }}}\right)}^2}dt}$

Parameter Estimation

Maximum Likelihood Estimation Method

The complete Eyring-lognormal log-likelihood function is composed of two summation portions:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\underset{i=1}{\sum ^{F_e}}{N}_i\ln \left[\frac{1}{{\sigma }_{T^{\prime }}{T}_i}\varphi \left(\frac{\ln \left(T_i\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)\right]\\ & & +\underset{i=1}{\sum ^S}{N}_i^{\prime }\ln [1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)]\\ & & +\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\ln [\mathrm{\Phi }(z_{Ri}^{\prime \prime })-\mathrm{\Phi }(z_{Li}^{\prime \prime })]\end{array}$

where:

$z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }+\ln V_i+A-\frac{B}{V_i}}{{\sigma }_T^{\prime }}$

$z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }+\ln V_i+A-\frac{B}{V_i}}{{\sigma }_T^{\prime }}$

and:
• $F_e$ is the number of groups of exact times-to-failure data points.
• $N_i$ is the number of times-to-failure data points in the $i^{th}$ time-to-failure data group.
• ${\sigma }_{T^{\prime }}$ is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).
• $A$ is the Eyring parameter (unknown, the second of three parameters to be estimated).
• $C$ is the second Eyring parameter (unknown, the third of three parameters to be estimated).
• $V_i$ is the stress level of the $i^{th}$ group.
• $T_i$ is the exact failure time of the $i^{th}$ group.
• $S$ is the number of groups of suspension data points.
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points.
• $T_i^{\prime }$ is the running time of the $i^{th}$ suspension data group.
• $FI$ is the number of interval data groups.
• $N_i^{\prime \prime }$ is the number of intervals in the i ${}^{th}$ group of data intervals.
• $T_{Li}^{\prime \prime }$ is the beginning of the i ${}^{th}$ interval.
• $T_{Ri}^{\prime \prime }$ is the ending of the i ${}^{th}$ interval. The solution (parameter estimates) will be found by solving for ${\hat{\sigma }}_{T^{\prime }},$ $\hat{A},$ $\hat{B}$ so that $\frac{\partial \mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial A}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial B}=0$ :

$\begin{array}{rlr}& \frac{\partial \mathrm{\Lambda }}{\partial A}=& -\frac{1}{{\sigma }_{T^{\prime }}^2}\underset{i=1}{\sum ^{F_e}}{N}_i(\ln (T_i)+\ln (V_i)+A-\frac{B}{V_i})\\ & & -\frac{1}{{\sigma }_{T^{\prime }}}\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{\varphi \left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)}{1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)}\\ & & \stackrel{FI}{+\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\frac{\phi (z_{Ri}^{\prime \prime })-\phi (z_{Li}^{\prime \prime })}{{\sigma }_T^{\prime }(\mathrm{\Phi }(z_{Ri}^{\prime \prime })-\mathrm{\Phi }(z_{Li}^{\prime \prime }))}\end{array}$

$\begin{array}{rlr}& \frac{\partial \mathrm{\Lambda }}{\partial B}=& \frac{1}{{\sigma }_{T^{\prime }}^2}\underset{i=1}{\sum ^{F_e}}{N}_i\frac{1}{V_i}(\ln (T_i)+\ln (V_i)+A-\frac{B}{V_i})\\ & & +\frac{1}{{\sigma }_{T^{\prime }}}\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{1}{V_i}\frac{\varphi \left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)}{1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)}\\ & & \stackrel{FI}{\underset{i=1}{-\sum ^{}_{}}}{N}_i^{\prime \prime }\frac{\phi (z_{Ri}^{\prime \prime })-\phi (z_{Li}^{\prime \prime })}{{\sigma }_T^{\prime }{V}_i(\mathrm{\Phi }(z_{Ri}^{\prime \prime })-\mathrm{\Phi }(z_{Li}^{\prime \prime }))}\\ & \frac{\partial \mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}}=& \underset{i=1}{\sum ^{F_e}}{N}_i(\frac{{(\ln (T_i)+\ln (V_i)+A-\frac{B}{V_i})}^2}{{\sigma }_{T^{\prime }}^3}-\frac{1}{{\sigma }_{T^{\prime }}})\\ & & +\frac{1}{{\sigma }_{T^{\prime }}}\underset{i=1}{\sum ^S}{N}_i^{\prime }\frac{\left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)\varphi \left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)}{1-\mathrm{\Phi }\left(\frac{\ln \left(T_i^{\prime }\right)+\ln (V_i)+A-\frac{B}{V_i}}{{\sigma }_{T^{\prime }}}\right)}\\ & & \stackrel{FI}{\underset{i=1}{-\sum ^{}_{}}}{N}_i^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\phi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\phi (z_{Li}^{\prime \prime })}{{\sigma }_T^{\prime }(\mathrm{\Phi }(z_{Ri}^{\prime \prime })-\mathrm{\Phi }(z_{Li}^{\prime \prime }))}\end{array}$

and:

$\varphi \left(x\right)=\frac{1}{\sqrt{2\pi }}\cdot e^{-\frac{1}{2}{\left(x\right)}^2}$

$\mathrm{\Phi }(x)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{x}{}}{e}^{-\frac{t^2}{2}}dt$

Generalized Eyring Relationship

Introduction

The generalized Eyring relationship is used when temperature and a second non-thermal stress (e.g. voltage) are the accelerated stresses of a test and their interaction is also of interest. This relationship is given by:

$L(V,U)=\frac{1}{V}{e}^{A+\frac{B}{V}+CU+D\frac{U}{V}}$

where:

• is the temperature (in absolute units ).

• $U$ is the non-thermal stress (i.e. voltage, vibration, etc.). $A,B,C,D$ are the parameters to be determined.

The Eyring relationship is a simple case of the generalized Eyring relationship where $C=D=0$ and $A_{Eyr}=-A_{GEyr}.$ Note that the generalized Eyring relationship includes the interaction term of $U$ and $V$ as described by the $D\frac{U}{V}$ term. In other words, this model can estimate the effect of changing one of the factors depending on the level of the other factor.

Acceleration Factor

Most models in actual use do not include any interaction terms, therefore, the acceleration factor can be computed by multiplying the acceleration factors obtained by changing each factor while keeping the other factors constant. In the case of the generalized Eyring relationship, the acceleration factor is derived differently.

The acceleration factor for the generalized Eyring relationship is given by:

$\begin{array}{rlr}& A_F=& \frac{L_{USE}}{L_{Accelerated}}=\frac{\frac{1}{V_U}{e}^{A+\frac{B}{V_U}+C{U}_U+D\frac{U_U}{V_U}}}{\frac{1}{T_A}{e}^{A+\frac{B}{V_A}+C{U}_A+D\frac{U_A}{V_A}}}\\ & =& \frac{\frac{1}{V_U}{e}^{A+\frac{B}{V_U}+C{U}_U+D\frac{U_U}{V_U}}}{\frac{1}{V_A}{e}^{A+\frac{B}{V_A}+C{U}_A+D\frac{U_A}{V_A}}}\end{array}$

where:
• $L_{USE}$ is the life at use stress level.

• $L_{Accelerated}$ is the life at the accelerated stress level.

• $V_u$ is the use temperature level.

• $V_A$ is the accelerated temperature level.

• $U_A$ is the accelerated non-thermal level.

• $U_u$ is the use non-thermal level.

Generalized Eyring-Exponential

By setting $m=L(V,U)$ as given in Eqn. (Gen-Eyr), the exponential $pdf$ becomes:

$f(t,V,U)=\left(V{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right)e^{-tV{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}}$

Generalized Eyring-Exponential Reliability Function

The generalized Eyring exponential model reliability function is given by:

$R(T,U,V)=e^{-tV{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}}$

Parameter Estimation

Substituting the generalized Eyring relationship into the exponential log-likelihood equation yields:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\stackrel{Fe}{\sum ^{}_{}_{i=1}}{N}_i\ln \left(V_i{e}^{-A-\frac{B}{V_i}-C{U}_i-D\frac{U_i}{V_i}}\right)\\ & & \stackrel{Fe}{\underset{i=1}{-\sum ^{}_{}}}{N}_i\left(T_i{V}_i{e}^{-A-\frac{B}{V_i}-C{U}_i-D\frac{U_i}{V_i}}\right)\\ & & -\stackrel{S}{\sum ^{}_{}_{i=1}}{N}_i^{\prime }\left(T_i^{\prime }{V}_i^{\prime }{e}^{-A-\frac{B}{V_i^{\prime }}-C{U}_i^{\prime }-D\frac{U_i^{\prime }}{V_i^{\prime }}}\right)\\ & & +\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]\end{array}$

where:

$R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime })=e^{-T_{Li}^{\prime \prime }{V}_i^{\prime \prime }{e}^{-A-\frac{B}{V_i^{\prime \prime }}-C{U}_i-D\frac{U_i}{V_i^{\prime \prime }}}}$

$R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime })=e^{-T_{Ri}^{\prime \prime }{V}_i^{\prime \prime }{e}^{-A-\frac{B}{V_i^{\prime \prime }}-C{U}_i-D\frac{U_i}{V_i^{\prime \prime }}}}$

and:
• $F_e$ is the number of groups of exact times-to-failure data points.
• $N_i$ is the number of times-to-failure data points in the $i^{th}$ time-to-failure data group.
• $A,B,C,D$ are parameters to be estimated.
• $V_i$ is the temperature level of the $i^{th}$ group.
• $U_i$ is the non-thermal stress level of the $i^{th}$ group.
• $T_i$ is the exact failure time of the $i^{th}$ group.
• $S$ is the number of groups of suspension data points.
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points.
• $T_i^{\prime }$ is the running time of the $i^{th}$ suspension data group.
• $FI$ is the number of interval data groups.
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals.
• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval.
• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval.
The solution (parameter estimates) will be found by solving for the parameters $A,$ $B,$ $C,$ and $D$ so that $\frac{\partial \mathrm{\Lambda }}{\partial A}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial B}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial D}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial D}=0$ .

Generalized Eyring-Weibull

By setting $\eta =L(V,U)$ from Eqn. (Gen-Eyr), the generalized Eyring Weibull model is given by:

$\begin{array}{rlr}& f(t,V,U)=& \beta \left(V{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right){\left(tV{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right)}^{\beta -1}\\ & & .e^{-{\left(tV{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right)}^{\beta }}\end{array}$

Generalized Eyring-Weibull Reliability Function

The generalized Eyring Weibull reliability function is given by:

$R(T,V,U)=e^{-{\left(tV{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right)}^{\beta }}$

Parameter Estimation

Substituting the generalized Eyring model into the Weibull log-likelihood equation yields:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\stackrel{Fe}{\sum ^{}_{}_{i=1}}{N}_i\ln [\beta \left(V{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right)\\ & & {\left(tV{e}^{-A-\frac{B}{V}-CU-D\frac{U}{V}}\right)}^{\beta -1}]\\ & & -\stackrel{Fe}{\sum ^{}_{}_{i=1}}{N}_i{\left(t_i{V}_i{e}^{-A-\frac{B}{V_i}-C{U}_i-D\frac{U_i}{V_i}}\right)}^{\beta }\\ & & -\stackrel{S}{\sum ^{}_{}_{i=1}}{N}_i^{\prime }{\left(t_i^{\prime }{V}_i^{\prime }{e}^{-A-\frac{B}{V_i^{\prime }}-C{U}_i^{\prime }-D\frac{U_i^{\prime }}{V_i^{\prime }}}\right)}^{\beta }\\ & & +\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]\end{array}$

where:

$R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime })=e^{-{\left(T_{Li}^{\prime \prime }{V}_i^{\prime \prime }{e}^{-A-\frac{B}{V_i^{\prime \prime }}-C{U}_i-D\frac{U_i}{V_i^{\prime \prime }}}\right)}^{\beta }}$

$R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime })=e^{-{\left(T_{Ri}^{\prime \prime }{V}_i^{\prime \prime }{e}^{-A-\frac{B}{V_i^{\prime \prime }}-C{U}_i-D\frac{U_i}{V_i^{\prime \prime }}}\right)}^{\beta }}$

and:
• $F_e$ is the number of groups of exact times-to-failure data points.

• $N_i$ is the number of times-to-failure data points in the $i^{th}$ time-to-failure data group.

• $A,B,C,D$ are parameters to be estimated.

• $V_i$ is the temperature level of the $i^{th}$ group.

• $U_i$ is the non-thermal stress level of the $i^{th}$ group.

• $T_i$ is the exact failure time of the $i^{th}$ group.

• $S$ is the number of groups of suspension data points.

• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points.

• $T_i^{\prime }$ is the running time of the $i^{th}$ suspension data group.

• $FI$ is the number of interval data groups.

• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals.

• $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval.

• $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval.

The solution (parameter estimates) will be found by solving for the parameters $A,$ $B,$ $C,$ and $D$ so that $\frac{\partial \mathrm{\Lambda }}{\partial A}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial B}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial D}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial D}=0$ .

Generalized Eyring-Lognormal

By setting ${\sigma }_T^{\prime }=L(V,U)$ from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by:

$f(t,V,U)=\frac{\phi (z(t))}{{\sigma }_T^{\prime }t}$

where:

$$

Generalized Eyring-Lognormal Reliability Function

The generalized Erying lognormal reliability function is given by:

$R(T,V,U)=1-\mathrm{\Phi }(z)$

Parameter Estimation

Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:

$\begin{array}{rlr}& \ln (L)=& \mathrm{\Lambda }=\stackrel{Fe}{\sum ^{}_{}_{i=1}}{N}_i\ln [\frac{\phi (z(t))}{{\sigma }_T^{\prime }t}]\stackrel{S}{\underset{i=1}{+\sum ^{}_{}}}{N}_i^{\prime }\ln (1-\mathrm{\Phi }(z(t_i^{\prime })))\\ & & +\stackrel{FI}{\sum ^{}_{}_{i=1}}{N}_i^{\prime \prime }\ln [\mathrm{\Phi }(z_{Ri}^{\prime \prime })-\mathrm{\Phi }(z_{Li}^{\prime \prime })]\end{array}$

where:

$z_{Ri}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\frac{B}{V_i^{\prime \prime }}-C{U}_i-D\frac{U_i}{V_i^{\prime \prime }}+\ln (V_i^{\prime \prime })}{{\sigma }_T^{\prime }}$

$z_{Li}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\frac{B}{V_i^{\prime \prime }}-C{U}_i-D\frac{U_i}{V_i^{\prime \prime }}+\ln (V_i^{\prime \prime })}{{\sigma }_T^{\prime }}$

and:
• $F_e$ is the number of groups of exact times-to-failure data points.
• $N_i$ is the number of times-to-failure data points in the $i^{th}$ time-to-failure data group.
• $A,B,C,D$ are parameters to be estimated.
• $V_i$ is the temperature level of the $i^{th}$ group.
• $U_i$ is the non-thermal stress level of the $i^{th}$ group.
• $T_i$ is the exact failure time of the $i^{th}$ group.
• $S$ is the number of groups of suspension data points.
• $N_i^{\prime }$ is the number of suspensions in the $i^{th}$ group of suspension data points. • $T_i^{\prime }$ is the running time of the $i^{th}$ suspension data group. • $FI$ is the number of interval data groups.
• $N_i^{\prime \prime }$ is the number of intervals in the $i^{th}$ group of data intervals. • $T_{Li}^{\prime \prime }$ is the beginning of the $i^{th}$ interval. • $T_{Ri}^{\prime \prime }$ is the ending of the $i^{th}$ interval.

The solution (parameter estimates) will be found by solving for the parameters $A,$ $B,$ $C,$ and $D$ so that $\frac{\partial \mathrm{\Lambda }}{\partial A}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial B}=0,$ $\frac{\partial \mathrm{\Lambda }}{\partial D}=0$ and $\frac{\partial \mathrm{\Lambda }}{\partial D}=0$ .

Example

The following data set represents failure times (in hours) obtained from an electronics epoxy packaging accelerated life test performed to understand the synergy between temperature and humidity and estimate the $B10$ life at the use conditions of $T=350K$ and $H=0.3$ . The data set is modeled using the lognormal distribution and the generalized Eyring model.

$$

The probability plot at the use conditions is shown next.

The $B10$ information is estimated to be 3004.63 hours, as shown next.

Appendix 7A: Eyring Confidence Bounds

Approximate Confidence Bounds for the Eyring-Exponential

Confidence Bounds on Mean Life

The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting $m=L(V)$ . The upper $(m_U)$ and lower $(m_L)$ bounds on the mean life (ML estimate of the mean life) are estimated by:

$m_U=\hat{m}\cdot e^{\frac{K_{\alpha }\sqrt{Var(\hat{m})}}{\hat{m}}}$

$m_L=\hat{m}\cdot e^{-\frac{K_{\alpha }\sqrt{Var(\hat{m})}}{\hat{m}}}$

where $K_{\alpha }$ is defined by:

$\alpha =\frac{1}{\sqrt{2\pi }}\underset{K_{\alpha }}{\overset{\mathrm{\infty }}{}}{e}^{-\frac{t^2}{2}}dt=1-\mathrm{\Phi }(K_{\alpha })$

If $\delta$ is the confidence level, then $\alpha =\frac{1-\delta }{2}$ for the two-sided bounds, and $\alpha =1-\delta$ for the one-sided bounds. The variance of $\hat{m}$ is given by:

$\begin{array}{rlr}& Var(\hat{m})=& {\left(\frac{\partial m}{\partial A}\right)}^{2}Var(\hat{A})+{\left(\frac{\partial m}{\partial B}\right)}^{2}Var(\hat{B})\\ & & +2\left(\frac{\partial m}{\partial A}\right)\left(\frac{\partial m}{\partial B}\right)Cov(\hat{A},\hat{B})\end{array}$

or:

$Var(\hat{m})=\frac{1}{V^2}{e}^{-2(\hat{A}-\frac{\hat{B}}{V})}[Var(\hat{A})+\frac{1}{V^2}Var(\hat{B})-\frac{1}{V}Cov(\hat{A},\hat{B})]$

The variances and covariance of $A$ and $B$ are estimated from the local Fisher matrix (evaluated at $\hat{A}$ , $\hat{B})$ as follows:

$\left[\begin{array}{cc}Var(\hat{A})& Cov(\hat{A},\hat{B})\\ Cov(\hat{B},\hat{A})& Var(\hat{B})\end{array}\right]={\left[\begin{array}{cc}-\frac{{\partial }^2\mathrm{\Lambda }}{\partial A^2}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A\partial B}\\ -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B\partial A}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B^2}\end{array}\right]}^{-1}$

Confidence Bounds on Reliability

The bounds on reliability at a given time, $T$ , are estimated by:

$\begin{array}{rlr}& R_U=& e^{-\frac{T}{m_U}}\\ & & \\ & R_L=& e^{-\frac{T}{m_L}}\end{array}$

where $m_U$ and $m_L$ are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).

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:

$\hat{T}=-\hat{m}\cdot \ln (R)$

The corresponding confidence bounds are estimated from:

$\begin{array}{rlr}& T_U=& -m_U\cdot \ln (R)\\ & & \\ & T_L=& -m_L\cdot \ln (R)\end{array}$

where $m_U$ and $m_L$ are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).

Approximate Confidence Bounds for the Eyring-Weibull

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since $\hat{\beta }$ is a positive parameter, $\ln (\hat{\beta })$ can then be treated as normally distributed. After performing this transformation, the bounds on the parameters are estimated from:

$\begin{array}{rlr}& {\beta }_U=& \hat{\beta }\cdot e^{\frac{K_{\alpha }\sqrt{Var(\hat{\beta })}}{\hat{\beta }}}\\ & {\beta }_L=& \hat{\beta }\cdot e^{-\frac{K_{\alpha }\sqrt{Var(\hat{\beta })}}{\hat{\beta }}}\end{array}$

also:

$\begin{array}{rlr}& A_U=& \hat{A}+K_{\alpha }\sqrt{Var(\hat{A})}\\ & A_L=& \hat{A}-K_{\alpha }\sqrt{Var(\hat{A})}\end{array}$

and:

$\begin{array}{rlr}& B_U=& \hat{B}+K_{\alpha }\sqrt{Var(\hat{B})}\\ & B_L=& \hat{B}-K_{\alpha }\sqrt{Var(\hat{B})}\end{array}$

The variances and covariances of $\beta ,$ $A,$ and $B$ are estimated from the Fisher matrix (evaluated at $\hat{\beta },$ $\hat{A},$ $\hat{B})$ as follows:

$\left[\begin{array}{ccc}Var(\hat{\beta })& Cov(\hat{\beta },\hat{A})& Cov(\hat{\beta },\hat{B})\\ Cov(\hat{A},\hat{\beta })& Var(\hat{A})& Cov(\hat{A},\hat{B})\\ Cov(\hat{B},\hat{\beta })& Cov(\hat{B},\hat{A})& Var(\hat{B})\end{array}\right]={\left[\begin{array}{ccc}-\frac{{\partial }^2\mathrm{\Lambda }}{\partial {\beta }^2}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial \beta \partial A}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial \beta \partial B}\\ -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A\partial \beta }& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A^2}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A\partial B}\\ -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B\partial \beta }& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B\partial A}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B^2}\end{array}\right]}^{-1}$

Confidence Bounds on Reliability

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

$\hat{R}(T,V)=e^{-{(T\cdot V\cdot e^{(\hat{A}-\frac{\hat{B}}{V})})}^{\hat{\beta }}}$

or:

$\hat{R}(T,V)=e^{-e^{\ln \left[{(T\cdot V\cdot e^{(\hat{A}-\frac{\hat{B}}{V})})}^{\hat{\beta }}\right]}}$

Setting:

$\hat{u}=\ln \left[{(T\cdot V\cdot e^{(\hat{A}-\frac{\hat{B}}{V})})}^{\hat{\beta }}\right]$

or:

$\hat{u}=\hat{\beta }[\ln (T)+\ln (V)+\hat{A}-\frac{\hat{B}}{V}]$

The reliability function now becomes:

$\hat{R}(T,V)=e^{-e\widehat{{}^u}}$

The next step is to find the upper and lower bounds on $\hat{u}$ :

$u_U=\hat{u}+K_{\alpha }\sqrt{Var(\hat{u})}$

$u_L=\hat{u}-K_{\alpha }\sqrt{Var(\hat{u})}$

where:

$\begin{array}{rlr}& Var(\hat{u})=& {\left(\frac{\partial \hat{u}}{\partial \beta }\right)}^{2}Var(\hat{\beta })+{\left(\frac{\partial \hat{u}}{\partial A}\right)}^{2}Var(\hat{A})\\ & & +{\left(\frac{\partial \hat{u}}{\partial B}\right)}^{2}Var(\hat{B})\\ & & +2\left(\frac{\partial \hat{u}}{\partial \beta }\right)\left(\frac{\partial \hat{u}}{\partial A}\right)Cov(\hat{\beta },\hat{A})\\ & & +2\left(\frac{\partial \hat{u}}{\partial \beta }\right)\left(\frac{\partial \hat{u}}{\partial B}\right)Cov(\hat{\beta },\hat{B})\\ & & +2\left(\frac{\partial \hat{u}}{\partial A}\right)\left(\frac{\partial \hat{u}}{\partial B}\right)Cov(\hat{A},\hat{B})\end{array}$

or:

$\begin{array}{rlr}& Var(\hat{u})=& {\left(\frac{\hat{u}}{\hat{\beta }}\right)}^{2}Var(\hat{\beta })+{\hat{\beta }}^{2}Var(\hat{A})\\ & & +{\left(\frac{\hat{\beta }}{V}\right)}^{2}Var(\hat{B})\\ & & +2\hat{u}\cdot Cov(\hat{\beta },\hat{A})-\frac{2\hat{u}}{V}Cov(\hat{\beta },\hat{B})\\ & & -\frac{2{\hat{\beta }}^2}{V}Cov(\hat{A},\hat{B})\end{array}$

The upper and lower bounds on reliability are:

$\begin{array}{rlr}& R_U=& e^{-e^{\left(u_L\right)}}\\ & R_L=& e^{-e^{\left(u_U\right)}}\end{array}$

where $u_U$ and $u_L$ are estimated using Eqns (EyrExpu) and (EyrExpl).

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:

$\begin{array}{rlr}& \ln (R)=& -{(\hat{T}\cdot V\cdot e^{(\hat{A}-\frac{\hat{B}}{V})})}^{\hat{\beta }}\\ & \ln (-\ln (R))=& \hat{\beta }(\ln \hat{T}+\ln V+\hat{A}-\frac{\hat{B}}{V})\end{array}$

or:

$\hat{u}=\frac{1}{\hat{\beta }}\ln (-\ln (R))-\ln V-\hat{A}+\frac{\hat{B}}{V}$

where

$$

The upper and lower bounds on $\hat{u}$ are then estimated from:

$u_U=\hat{u}+K_{\alpha }\sqrt{Var(\hat{u})}$

$u_L=\hat{u}-K_{\alpha }\sqrt{Var(\hat{u})}$

where:

$\begin{array}{rlr}& Var(\hat{u})=& {\left(\frac{\partial \hat{u}}{\partial \beta }\right)}^{2}Var(\hat{\beta })+{\left(\frac{\partial \hat{u}}{\partial A}\right)}^{2}Var(\hat{A})\\ & & +{\left(\frac{\partial \hat{u}}{\partial B}\right)}^{2}Var(\hat{B})\\ & & +2\left(\frac{\partial \hat{u}}{\partial \beta }\right)\left(\frac{\partial \hat{u}}{\partial A}\right)Cov(\hat{\beta },\hat{A})\\ & & +2\left(\frac{\partial \hat{u}}{\partial \beta }\right)\left(\frac{\partial \hat{u}}{\partial B}\right)Cov(\hat{\beta },\hat{B})\\ & & +2\left(\frac{\partial \hat{u}}{\partial A}\right)\left(\frac{\partial \hat{u}}{\partial B}\right)Cov(\hat{A},\hat{B})\end{array}$

or:

$\begin{array}{rlr}& Var(\hat{u})=& \frac{1}{{\hat{\beta }}^4}{[\ln (-\ln (R))]}^{2}Var(\hat{\beta })\\ & & +Var(\hat{A})+\frac{1}{V^2}Var(\hat{B})\\ & & +\frac{2\ln (-\ln (R))}{{\hat{\beta }}^2}Cov(\hat{\beta },\hat{A})-\frac{2\ln (-\ln (R))}{{\hat{\beta }}^{2}V}Cov(\hat{\beta },\hat{B})\\ & & -\frac{2}{V}Cov(\hat{A},\hat{B})\end{array}$

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

$\begin{array}{rlr}& T_U=& e^{u_U}\\ & T_L=& e^{u_L}\end{array}$

where $u_U$ and $u_L$ are estimated using Eqns. (EyrTimeu) and (EyrTimel).

Approximate Confidence Bounds for the Eyring-Lognormal

Bounds on the Parameters

The lower and upper bounds on $A$ and $B$ are estimated from:

$\begin{array}{rlr}& A_U=& \hat{A}+K_{\alpha }\sqrt{Var(\hat{A})}\text{ (Upper bound)}\\ & A_L=& \hat{A}-K_{\alpha }\sqrt{Var(\hat{A})}\text{ (Lower bound)}\end{array}$

and:

$\begin{array}{rlr}& B_U=& \hat{B}+K_{\alpha }\sqrt{Var(\hat{B})}\text{ (Upper bound)}\\ & B_L=& \hat{B}-K_{\alpha }\sqrt{Var(\hat{B})}\text{ (Lower bound)}\end{array}$

Since the standard deviation, ${\hat{\sigma }}_{T^{\prime },}$ is a positive parameter, $\ln ({\hat{\sigma }}_{T^{\prime }})$ is treated as normally distributed, and the bounds are estimated from:

$\begin{array}{rlr}& {\sigma }_U=& {\hat{\sigma }}_{T^{\prime }}\cdot e^{\frac{K_{\alpha }\sqrt{Var({\hat{\sigma }}_{T^{\prime }})}}{{\hat{\sigma }}_{T^{\prime }}}}\text{ (Upper bound)}\\ & {\sigma }_L=& \frac{{\hat{\sigma }}_{T^{\prime }}}{e^{\frac{K_{\alpha }\sqrt{Var({\hat{\sigma }}_{T^{\prime }})}}{{\hat{\sigma }}_{T^{\prime }}}}}\text{ (Lower bound)}\end{array}$

The variances and covariances of $A,$ $B,$ and ${\sigma }_{T^{\prime }}$ are estimated from the local Fisher matrix (evaluated at $\hat{A},$ $\hat{B}$ , ${\hat{\sigma }}_{T^{\prime }})$ as follows:

$\left(\begin{array}{ccc}Var\left({\hat{\sigma }}_{T^{\prime }}\right)& Cov(\hat{A},{\hat{\sigma }}_{T^{\prime }})& Cov(\hat{B},{\hat{\sigma }}_{T^{\prime }})\\ Cov({\hat{\sigma }}_{T^{\prime }},\hat{A})& Var\left(\hat{A}\right)& Cov(\hat{A},\hat{B})\\ Cov({\hat{\sigma }}_{T^{\prime }},\hat{B})& Cov(\hat{B},\hat{A})& Var\left(\hat{B}\right)\end{array}\right)={[F]}^{-1}$

where:

$F=\left(\begin{array}{ccc}-\frac{{\partial }^2\mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}^2}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}\partial A}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial {\sigma }_{T^{\prime }}\partial B}\\ -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A\partial {\sigma }_{T^{\prime }}}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A^2}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial A\partial B}\\ -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B\partial {\sigma }_{T^{\prime }}}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B\partial A}& -\frac{{\partial }^2\mathrm{\Lambda }}{\partial B^2}\end{array}\right)$

Bounds on Reliability

The reliability of the lognormal distribution is given by:

$R(T^{\prime },V;A,B,{\sigma }_{T^{\prime }})=\underset{T^{\prime }}{\overset{\mathrm{\infty }}{}}\frac{1}{{\hat{\sigma }}_{T^{\prime }}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{t+\ln (V)+\hat{A}-\frac{\hat{B}}{V}}{{\hat{\sigma }}_{T^{\prime }}}\right)}^2}dt$

Let $\hat{z}(t,V;A,B,{\sigma }_T)=\frac{t+\ln (V)+\hat{A}-\frac{\hat{B}}{V}}{{\hat{\sigma }}_{T^{\prime }}},$ then $\frac{d\hat{z}}{dt}=\frac{1}{{\hat{\sigma }}_{T^{\prime }}}.$
For $t=T^{\prime }$ , $\hat{z}=\frac{T^{\prime }+\ln (V)+\hat{A}-\frac{\hat{B}}{V}}{{\hat{\sigma }}_{T^{\prime }}}$ , and for $t=\mathrm{\infty },$ $\hat{z}=\mathrm{\infty }.$ The above equation then becomes:

$R(\hat{z})=\underset{\hat{z}(T^{\prime },V)}{\overset{\mathrm{\infty }}{}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{z}^2}dz$

The bounds on $z$ are estimated from:

$\begin{array}{rlr}& z_U=& \hat{z}+K_{\alpha }\sqrt{Var(\hat{z})}\\ & z_L=& \hat{z}-K_{\alpha }\sqrt{Var(\hat{z})}\end{array}$

where:

$\begin{array}{rlr}& Var(\hat{z})=& {\left(\frac{\partial \hat{z}}{\partial A}\right)}_{\hat{A}}^{2}Var(\hat{A})+{\left(\frac{\partial \hat{z}}{\partial B}\right)}_{\hat{B}}^{2}Var(\hat{B})+{\left(\frac{\partial \hat{z}}{\partial {\sigma }_{T^{\prime }}}\right)}_{{\hat{\sigma }}_{T^{\prime }}}^{2}Var({\hat{\sigma }}_T)\\ & & +2{\left(\frac{\partial \hat{z}}{\partial A}\right)}_{\hat{A}}{\left(\frac{\partial \hat{z}}{\partial B}\right)}_{\hat{B}}Cov(\hat{A},\hat{B})\\ & & +2{\left(\frac{\partial \hat{z}}{\partial A}\right)}_{\hat{A}}{\left(\frac{\partial \hat{z}}{\partial {\sigma }_{T^{\prime }}}\right)}_{{\hat{\sigma }}_{T^{\prime }}}Cov(\hat{A},{\hat{\sigma }}_T)\\ & & +2{\left(\frac{\partial \hat{z}}{\partial B}\right)}_{\hat{B}}{\left(\frac{\partial \hat{z}}{\partial {\sigma }_{T^{\prime }}}\right)}_{{\hat{\sigma }}_{T^{\prime }}}Cov(\hat{B},{\hat{\sigma }}_T)\end{array}$

or:

$\begin{array}{rlr}& Var(\hat{z})=& \frac{1}{{\hat{\sigma }}_{T^{\prime }}^2}[Var(\hat{A})+\frac{1}{V^2}Var(\hat{B})+{\hat{z}}^{2}Var({\hat{\sigma }}_{T^{\prime }})\\ & & -\frac{2}{V}Cov(\hat{A},\hat{B})-2\hat{z}Cov(\hat{A},{\hat{\sigma }}_{T^{\prime }})+\frac{2\hat{z}}{V}Cov(\hat{B},{\hat{\sigma }}_{T^{\prime }})]\end{array}$

The upper and lower bounds on reliability are:

$\begin{array}{rlr}& R_U=& \underset{z_L}{\overset{\mathrm{\infty }}{}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{z}^2}dz\text{ (Upper bound)}\\ & R_L=& \underset{z_U}{\overset{\mathrm{\infty }}{}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{z}^2}dz\text{ (Lower bound)}\end{array}$

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:

$T^{\prime }(V;\hat{A},\hat{B},{\hat{\sigma }}_{T^{\prime }})=-\ln (V)-\hat{A}+\frac{\hat{B}}{V}+z\cdot {\hat{\sigma }}_{T^{\prime }}$

where:

$\begin{array}{rlr}& T^{\prime }(V;\hat{A},\hat{B},{\hat{\sigma }}_{T^{\prime }})=& \ln (T)\\ & z=& {\mathrm{\Phi }}^{-1}[F(T^{\prime })]\end{array}$

and:

$\mathrm{\Phi }(z)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{z(T^{\prime })}{}}{e}^{-\frac{1}{2}{z}^2}dz$

The next step is to calculate the variance of $T^{\prime }(V;\hat{A},\hat{B},{\hat{\sigma }}_{T^{\prime }}):$

$\begin{array}{rlr}& Var(T^{\prime })=& {\left(\frac{\partial T^{\prime }}{\partial A}\right)}^{2}Var(\hat{A})+{\left(\frac{\partial T^{\prime }}{\partial B}\right)}^{2}Var(\hat{B})+{\left(\frac{\partial T^{\prime }}{\partial {\sigma }_{T^{\prime }}}\right)}^{2}Var({\hat{\sigma }}_{T^{\prime }})\\ & & +2\left(\frac{\partial T^{\prime }}{\partial A}\right)\left(\frac{\partial T^{\prime }}{\partial B}\right)Cov(\hat{A},\hat{B})\\ & & +2\left(\frac{\partial T^{\prime }}{\partial A}\right)\left(\frac{\partial T^{\prime }}{\partial {\sigma }_{T^{\prime }}}\right)Cov(\hat{A},{\hat{\sigma }}_{T^{\prime }})\\ & & +2\left(\frac{\partial T^{\prime }}{\partial B}\right)\left(\frac{\partial T^{\prime }}{\partial {\sigma }_{T^{\prime }}}\right)Cov(\hat{B},{\hat{\sigma }}_{T^{\prime }})\end{array}$

or:

$\begin{array}{rlr}& Var(T^{\prime })=& Var(\hat{A})+\frac{1}{V}Var(\hat{B})+{\hat{z}}^{2}Var({\hat{\sigma }}_{T^{\prime }})\\ & & -\frac{2}{V}Cov(\hat{A},\hat{B})\\ & & -2\hat{z}Cov(\hat{A},{\hat{\sigma }}_{T^{\prime }})\\ & & +\frac{2\hat{z}}{V}Cov(\hat{B},{\hat{\sigma }}_{T^{\prime }})\end{array}$

The upper and lower bounds are then found by:

$\begin{array}{rlr}& T_U^{\prime }=& \ln T_U=T^{\prime }+K_{\alpha }\sqrt{Var(T^{\prime })}\\ & T_L^{\prime }=& \ln T_L=T^{\prime }-K_{\alpha }\sqrt{Var(T^{\prime })}\end{array}$

Solving for $T_U$ and $T_L$ yields:

$\begin{array}{rlr}& T_U=& e^{T_U^{\prime }}\text{ (Upper bound)}\\ & T_L=& e^{T_L^{\prime }}\text{ (Lower bound)}\end{array}$