Template:Gen-eyring log par est

Parameter Estimation

Substituting the generalized Eyring model into the lognormal 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 [\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}]\overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(t_{i}^{\prime })) \right) +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })] \end{align} }[/math]

where:

[math]\displaystyle{ z_{Ri}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} }[/math]

[math]\displaystyle{ z_{Li}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} }[/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] .