Template:Gen-eyring weib rf

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


 * $$\begin{align}

& \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\beta \left( V{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right) \\ & & {{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta -1}}] \\ & & -\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( {{t}_{i}}{{V}_{i}}{{e}^{-A-\tfrac{B}{{{V}_{i}}}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{{{V}_{i}}}}} \right)}^{\beta }} \\ & & -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( t_{i}^{\prime }V_{i}^{\prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime }}-CU_{i}^{\prime }-D\tfrac{U_{i}^{\prime }}{V_{i}^{\prime }}}} \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:


 * $$R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime })={{e}^{-{{\left( T_{Li}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}} \right)}^{\beta }}}}$$


 * $$R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime })={{e}^{-{{\left( T_{Ri}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}} \right)}^{\beta }}}}$$


 * 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  $$\tfrac{\partial \Lambda }{\partial A}=0,$$   $$\tfrac{\partial \Lambda }{\partial B}=0,$$   $$\tfrac{\partial \Lambda }{\partial D}=0$$  and  $$\tfrac{\partial \Lambda }{\partial D}=0$$.