Template:Gen-eyring weib rf

Parameter Estimation

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

[math]\displaystyle{ \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} }[/math]

where:

[math]\displaystyle{ 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 }}}} }[/math]

[math]\displaystyle{ 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 }}}} }[/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 data points in the [math]\displaystyle{ {{i}^{th}} }[/math] time-to-failure data group.

• [math]\displaystyle{ A,B,C,D }[/math] are parameters to be estimated.

• [math]\displaystyle{ {{V}_{i}} }[/math] is the temperature level of the [math]\displaystyle{ {{i}^{th}} }[/math] group.

• [math]\displaystyle{ {{U}_{i}} }[/math] is the non-thermal stress level of the [math]\displaystyle{ {{i}^{th}} }[/math] group.

• [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 [math]\displaystyle{ {{i}^{th}} }[/math] group of data intervals.

• [math]\displaystyle{ T_{Li}^{\prime \prime } }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.

• [math]\displaystyle{ T_{Ri}^{\prime \prime } }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.

The solution (parameter estimates) will be found by solving for the parameters [math]\displaystyle{ A, }[/math] [math]\displaystyle{ B, }[/math] [math]\displaystyle{ C, }[/math] and [math]\displaystyle{ D }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial A}=0, }[/math] [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial B}=0, }[/math] [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial D}=0 }[/math] and [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial D}=0 }[/math] .