Template:T-h lognormal

T-H Lognormal

The [math]\displaystyle{ pdf }[/math] of the lognormal distribution is given by:

[math]\displaystyle{ f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} }[/math]

where:

[math]\displaystyle{ {T}'=\ln (T) }[/math]

[math]\displaystyle{ T=\text{times-to-failure} }[/math]

and:
• [math]\displaystyle{ \overline{{{T}'}}= }[/math] mean of the natural logarithms of the times-to-failure.
• [math]\displaystyle{ {{\sigma }_{{{T}'}}}= }[/math] standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

[math]\displaystyle{ \breve{T}={{e}^{{{\overline{T}}^{\prime }}}} }[/math]

The T-H lognormal model [math]\displaystyle{ pdf }[/math] can be obtained first by setting [math]\displaystyle{ \breve{T} =L(V,U) }[/math].
Therefore:

[math]\displaystyle{ \breve{T}=L(V,U)=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}} }[/math]

or:

[math]\displaystyle{ {{e}^{{{\overline{T}}^{\prime }}}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}} }[/math]

Thus:

[math]\displaystyle{ {{\overline{T}}^{\prime }}=\ln (A)+\frac{\phi }{V}+\frac{b}{U}. }[/math]

Substituting the above equation into the lognormal [math]\displaystyle{ pdf }[/math] yields the T-H lognormal model [math]\displaystyle{ pdf }[/math] or:

[math]\displaystyle{ f(T,V,U)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} }[/math]

T-H Lognormal Statistical Properties Summary

The Mean

• The mean life of the T-H lognormal model (mean of the times-to-failure), [math]\displaystyle{ \bar{T} }[/math] , is given by:

[math]\displaystyle{ \begin{align} & \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} =\ {{e}^{\ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \end{align} }[/math]

• The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\bar{T}}^{^{\prime }}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:

[math]\displaystyle{ {{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right) }[/math]

The Standard Deviation

• The standard deviation of the T-H lognormal model (standard deviation of the times-to-failure), [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:

[math]\displaystyle{ \begin{align} & {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} =\ & \sqrt{\left( {{e}^{2\left( \ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \end{align} }[/math]

• The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma }_{T}} }[/math] is given by:

[math]\displaystyle{ {{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)} }[/math]

The Mode

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

[math]\displaystyle{ \begin{align} & \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}}=\ & {{e}^{\ln (A)+\tfrac{\phi }{V}+\tfrac{b}{U}-\sigma _{{{T}'}}^{2}}} \end{align} }[/math]

T-H Lognormal Reliability

The reliability for a mission of time [math]\displaystyle{ T }[/math] , starting at age 0, for the T-H lognormal model is determined by:

[math]\displaystyle{ R(T,V,U)=\int_{T}^{\infty }f(t,V,U)dt }[/math]

or:

[math]\displaystyle{ R(T,V,U)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

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 T-H lognormal model, the reliable life, or the mission duration for a desired reliability goal, [math]\displaystyle{ {{t}_{R}}, }[/math] is estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ T_{R}^{\prime }=\ln (A)+\frac{\phi }{V}+\frac{b}{U}+z\cdot {{\sigma }_{{{T}'}}} }[/math]

where:

[math]\displaystyle{ z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V,U \right) \right] }[/math]

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',V,U)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt }[/math]

Since [math]\displaystyle{ {T}'=\ln (T), }[/math] the reliable life, [math]\displaystyle{ {{t}_{R,}} }[/math] is given by:

[math]\displaystyle{ {{t}_{R}}={{e}^{T_{R}^{\prime }}} }[/math]

T-H Lognormal Failure Rate

The lognormal failure rate is given by:

[math]\displaystyle{ \lambda (T,V,U)=\frac{f(T,V,U)}{R(T,V,U)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt} }[/math]

Parameter Estimation

Maximum Likelihood Estimation Method

The complete T-H lognormal log-likelihood function is:

[math]\displaystyle{ \begin{align} & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}{{\phi }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (A)-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{{{U}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (A)-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{{{U}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })] \end{align} }[/math]

where:

[math]\displaystyle{ z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln A-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{U_{i}^{\prime \prime }}}{\sigma _{T}^{\prime }} }[/math]

[math]\displaystyle{ z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln A-\tfrac{\phi }{{{V}_{i}}}-\tfrac{b}{U_{i}^{\prime \prime }}}{\sigma _{T}^{\prime }} }[/math]

[math]\displaystyle{ {{\phi }_{pdf}}\left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}} }[/math]

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

and:
• [math]\displaystyle{ {{F}_{e}} }[/math] is the number of groups of exact times-to-failure data points.
• [math]\displaystyle{ {{N}_{i}} }[/math] is the number of times-to-failure data points in the [math]\displaystyle{ {{i}^{th}} }[/math] time-to-failure data group.
• [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of four parameters to be estimated).
• [math]\displaystyle{ A }[/math] is the first T-H parameter (unknown, the second of four parameters to be estimated).
• [math]\displaystyle{ \phi }[/math] is the second T-H parameter (unknown, the third of four parameters to be estimated).
• [math]\displaystyle{ b }[/math] is the third T-H parameter (unknown, the fourth of four parameters to be estimated).
• [math]\displaystyle{ {{V}_{i}} }[/math] is the stress level for the first stress type (i.e. temperature) of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ {{U}_{i}} }[/math] is the stress level for the second stress type (i.e. relative humidity) of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ {{T}_{i}} }[/math] is the exact failure time of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ S }[/math] is the number of groups of suspension data points.
• [math]\displaystyle{ N_{i}^{\prime } }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}} }[/math] group of suspension data points.
• [math]\displaystyle{ T_{i}^{\prime } }[/math] is the running time of the [math]\displaystyle{ {{i}^{th}} }[/math] suspension data group.
• [math]\displaystyle{ FI }[/math] is the number of interval data groups.
• [math]\displaystyle{ N_{i}^{\prime \prime } }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}} }[/math] group of data intervals.
• [math]\displaystyle{ T_{Li}^{\prime \prime } }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}} }[/math] interval. • [math]\displaystyle{ T_{Ri}^{\prime \prime } }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.
•
The solution (parameter estimates) will be found by solving for [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}, }[/math] [math]\displaystyle{ \widehat{A}, }[/math] [math]\displaystyle{ \widehat{\phi }, }[/math] [math]\displaystyle{ \widehat{b} }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0, }[/math] [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial A}=0, }[/math] [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \phi }=0 }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial b}=0 }[/math] .