Template:Eyring-weibull mle

Maximum Likelihood Estimation Method
The Eyring-Weibull log-likelihood function is composed of two summation portions:


 * $$\begin{align}

& \ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta \cdot {{V}_{i}}\cdot {{e}^{A-\tfrac{B}}}{{\left( {{T}_{i}}{{V}_{i}}{{e}^{A-\tfrac{B}}} \right)}^{\beta -1}}{{e}^{-{{\left( {{T}_{i}}{{V}_{i}}{{e}^{A-\tfrac{B}}} \right)}^{\beta }}}} \right] \\ & & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( {{V}_{i}}{{e}^{A-\tfrac{B}}}T_{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:


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


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


 * 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). •	 $$A$$ is the Eyring parameter (unknown, the second of three parameters to be estimated). •	 $$B$$ is the second Eyring 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 the parameters $$\beta ,$$   $$A$$  and  $$B$$  so that  $$\tfrac{\partial \Lambda }{\partial \beta }=0,$$   $$\tfrac{\partial \Lambda }{\partial A}=0$$  and  $$\tfrac{\partial \Lambda }{\partial B}=0$$
 * where:


 * $$\begin{align}

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


 * $$\frac{\partial \Lambda}{\partial \beta}=\frac{1}{\beta}\sum_{i=1}^{F_e} N_i\frac{1}{V_i}+\sum_{i=1}^{F_e} N_i ln\left(T_iV_i e^{A-\tfrac{B}{V_i}}\right)$$
 * $$-\sum_{i=1}^{F_e} N_i\left(T_iV_i e^{A-\tfrac{B}{V_i}}\right)^\beta ln\left(T_iV)i e^{A-\tfrac{B}{V_i}}\right)$$
 * $$-\sum_{i=1}^S N_i^'\left(T_i^'V_I e^{A-\tfrac{B}{V_i}}\right)^\beta ln\left(T_iV)i e^{A-\tfrac{B}{V_i}}\right)$$
 * $$-\sum_{i=1}^{FI} N_i^{}V_i e^{A-\tfrac{B}{V_i}}\frac{R_{Li}^{} T_{Li}^{}\left(ln(T_{Li}^' V_i)+A-\tfrac{B}{V_i}\right)-R_{Ri}^{} T_{Ri}^{}\left(ln(T_{Ri}^{} V_i)+A-\tfrac{B}{V_i}\right)}{R_{L_i}^{}-F_{Ri}^{}}$$