Template:Arrhenius-log mle

Maximum Likelihood Estimation Method
The lognormal log-likelihood function for the Arrhenius-lognormal model is as follows:


 * $$\begin{align}

& \ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)-\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)-\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:


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


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

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.

•	 $${{\sigma}_{{T}'}}$$ is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

•	 $$B$$ is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

•	 $$C$$ is the second Arrhenius 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 $${{\widehat{\sigma }}_},$$   $$\widehat{B},$$   $$\widehat{C}$$  so that  $$\tfrac{\partial \Lambda }{\partial {{\sigma }_}}=0,$$   $$\tfrac{\partial \Lambda }{\partial B}=0$$  and  $$\tfrac{\partial \Lambda }{\partial C}=0$$, where:
 * $$\begin{align}

& \frac{\partial \Lambda }{\partial B}= \frac{1}{\sigma _^{2}}\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\frac{1}(\ln ({{T}_{i}})-\ln (C)-\frac{B}) +\frac{1}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \end{align}$$


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial C}= \frac{1}{C\cdot \sigma _^{2}}\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}) +\frac{1}{C\cdot {{\sigma }_}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }C(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\ & &  \\  & \frac{\partial \Lambda }{\partial {{\sigma }_}}= \underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\left( \frac{\sigma _^{3}}-\frac{1} \right) +\frac{1}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \end{align}$$

and:


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


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