Template:TNT Lognormal

T-NT Lognormal
The $$pdf$$  of the lognormal distribution is given by:


 * $$f(T)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline}{{{\sigma }_}} \right)}^{2}}}}$$

where:


 * $${T}'=\ln (T)$$

and: •	 $$T=$$ times-to-failure. •	 $$\overline=$$ mean of the natural logarithms of the times-to-failure. •	 $${{\sigma }_}=$$ standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:


 * $$\breve{T}={{e}^}$$

The T-NT lognormal model $$pdf$$ can be obtained by setting $$\breve{T}=L(V)$$. Therefore:


 * $$\breve{T}=L(V)=\frac{C}{{e}^{\tfrac{B}{V}}}$$

or:


 * $${{e}^}=\frac{C}{{e}^{\tfrac{B}{V}}}$$

Thus:


 * $${{\overline{T}}^{\prime }}=\ln (C)-n\ln (U)+\frac{B}{V}$$

Substituting the above equation into the lognormal $$pdf$$ yields the T-NT lognormal model $$pdf$$ or:


 * $$f(T,U,V)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_}} \right)}^{2}}}}$$

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


 * $$\begin{align}

& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _^{2}}} = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}+\tfrac{1}{2}\sigma _^{2}}} \end{align}$$

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


 * $${{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}+1 \right)$$

The Standard Deviation
•	The standard deviation of the T-NT lognormal model (standard deviation of the times-to-failure), $${{\sigma }_{T}}$$, is given by:


 * $$\begin{align}

& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _^{2}}} \right)\left( {{e}^{\sigma _^{2}}}-1 \right)} = & \sqrt{\left( {{e}^{2\left( \ln (C)-n\ln (U)+\tfrac{B}{V} \right)+\sigma _^{2}}} \right)\left( {{e}^{\sigma _^{2}}}-1 \right)} \end{align}$$

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


 * $${{\sigma }_}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}+1 \right)}$$

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


 * $$\begin{align}

& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _^{2}}} = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}-\sigma _^{2}}} \end{align}$$

T-NT Lognormal Reliability
For the T-NT lognormal model, the reliability for a mission of time $$T$$, starting at age 0, for the T-NT lognormal model is determined by:


 * $$R(T,U,V)=\int_{T}^{\infty }f(t,U,V)dt$$

or:


 * $$R(T,U,V)=\int_^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt$$

Reliable Life
For the T-NT 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:


 * $$T_{R}^{\prime }=\ln (C)-n\ln (U)+\frac{B}{V}+z\cdot {{\sigma }_}$$

where:


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

and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{2}}}dt$$

Since $${T}'=\ln (T)$$  the reliable life,  $${{t}_{R}}$$, is given by:


 * $${{t}_{R}}={{e}^{T_{R}^{\prime }}}$$

Lognormal Failure Rate
The T-NT lognormal failure rate is given by:


 * $$\lambda (T,U,V)=\frac{f(T,U,V)}{R(T,U,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_}} \right)}^{2}}}}}{\int_^{\infty }\tfrac{1}{{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}$$

Maximum Likelihood Estimation Method
The complete T-NT lognormal log-likelihood function is:


 * $$\begin{align}

& \ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{\phi }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)+n\ln ({{U}_{i}})-\tfrac{B}} \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 (C)+n\ln ({{U}_{i}})-\tfrac{B}} \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}$$

where:


 * $$z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln C+n\ln U_{i}^{\prime \prime }-\tfrac{B}}{\sigma _{T}^{\prime }}$$


 * $$z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln C+n\ln U_{i}^{\prime \prime }-\tfrac{B}}{\sigma _{T}^{\prime }}$$


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


 * $$\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{2}}}dt$$

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. •	 $${{\sigma }_}$$ is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of four parameters to be estimated). •	 $$B$$ is the first T-NT parameter (unknown, the second of four parameters to be estimated). •	 $$C$$ is the second T-NT parameter (unknown, the third of four parameters to be estimated). •	 $$n$$ is the third T-NT parameter (unknown, the fourth of four parameters to be estimated). •	 $${{V}_{i}}$$ is the stress level for the first stress type (i.e. temperature) of the  $${{i}^{th}}$$  group. •	 $${{U}_{i}}$$ is the stress level for the second stress type (i.e. non-thermal) 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 $${{\widehat{\sigma }}_},$$   $$\widehat{B},$$   $$\widehat{C},$$   $$\widehat{n}$$  so that  $$\tfrac{\partial \Lambda }{\partial {{\sigma }_}}=0,$$   $$\tfrac{\partial \Lambda }{\partial B}=0,$$   $$\tfrac{\partial \Lambda }{\partial C}=0$$  and  $$\tfrac{\partial \Lambda }{\partial n}=0$$.