Template:Ipl ex mle

Maximum Likelihood Parameter Estimation

Substituting the inverse power law relationship into the exponential log-likelihood equation yields:

[math]\displaystyle{ \begin{align} & \ln (L)= \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ KV_{i}^{n}{{e}^{-KV_{i}^{n}{{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }KV_{i}^{n}T_{i}^{\prime }+\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} }[/math]

where:

[math]\displaystyle{ R_{Li}^{\prime \prime }={{e}^{-T_{Li}^{\prime \prime }KV_{i}^{n}}} }[/math]

[math]\displaystyle{ R_{Ri}^{\prime \prime }={{e}^{-T_{Ri}^{\prime \prime }KV_{i}^{n}}} }[/math]

and:
• [math]\displaystyle{ {{F}_{e}} }[/math] is the number of groups of exact times-to-failure data points.
• [math]\displaystyle{ {{N}_{i}} }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}} }[/math] time-to-failure data group.
• [math]\displaystyle{ {{V}_{i}} }[/math] is the stress level of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ K }[/math] is the IPL parameter (unknown, the first of two parameters to be estimated).
• [math]\displaystyle{ n }[/math] is the second IPL parameter (unknown, the second of two parameters to be estimated).
• [math]\displaystyle{ {{T}_{i}} }[/math] is the exact failure time of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ S }[/math] is the number of groups of suspension data points.
• [math]\displaystyle{ N_{i}^{\prime } }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}} }[/math] group of suspension data points.
• [math]\displaystyle{ T_{i}^{\prime } }[/math] is the running time of the [math]\displaystyle{ {{i}^{th}} }[/math] suspension data group.
• [math]\displaystyle{ FI }[/math] is the number of interval data groups.
• [math]\displaystyle{ N_{i}^{\prime \prime } }[/math] is the number of intervals in the i [math]\displaystyle{ ^{th} }[/math] group of data intervals.
• [math]\displaystyle{ T_{Li}^{\prime \prime } }[/math] is the beginning of the i [math]\displaystyle{ ^{th} }[/math] interval.
• [math]\displaystyle{ T_{Ri}^{\prime \prime } }[/math] is the ending of the i [math]\displaystyle{ ^{th} }[/math] interval.

The solution (parameter estimates) will be found by solving for the parameters [math]\displaystyle{ \widehat{K}, }[/math] [math]\displaystyle{ \widehat{n} }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial K}=0 }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial n}=0 }[/math] , where:

[math]\displaystyle{ \begin{align} & \frac{\partial \Lambda }{\partial K}= & \frac{1}{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}-\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}V_{i}^{n}{{T}_{i}}-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }V_{i}^{n}T_{i}^{\prime } \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\left( T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime } \right)V_{i}^{n}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \end{align} }[/math]

[math]\displaystyle{ \begin{align} & \frac{\partial \Lambda }{\partial n}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln ({{V}_{i}})-K\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}V_{i}^{n}\ln ({{V}_{i}}){{T}_{i}} -K\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }V_{i}^{n}\ln ({{V}_{i}})T_{i}^{\prime } \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{KV_{i}^{n}\ln ({{V}_{i}})\left( T_{Li}{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime } \right)}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \end{align} }[/math]