Template:Aaw mle

Maximum Likelihood Estimation Method
The Arrhenius-Weibull log-likelihood function is as follows:


 * $$\begin{align}

& \Lambda = & \underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}}}}{{\left( \frac{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta }}}} \right] \ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}}}} \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }] \end{align}$$

where:


 * $$R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)}^{\beta }}}}$$


 * $$R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}}}} \right)}^{\beta }}}}$$

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).

•	 $$B$$ 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 $$\widehat{\beta },$$   $$\widehat{B},$$   $$\widehat{C}$$  so that  $$\tfrac{\partial \Lambda }{\partial \beta }=0,$$   $$\tfrac{\partial \Lambda }{\partial B}=0$$  and  $$\tfrac{\partial \Lambda }{\partial C}=0$$, where:


 * $$\begin{align}

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


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial B}= -\beta \underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\frac{1}+\beta \underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}}}} \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} \end{align}$$


 * $$\begin{align}

& \frac{\partial \Lambda }{\partial C}= -\frac{\beta }{C}\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)} \end{align}$$