Inverse Power Law Relationship

=Appendix 8A: IPL Confidence Bounds=

Confidence Bounds on the Mean Life
From the inverse power law relationship the mean life for the exponential distribution is given by setting $$m=L(V)$$. The upper $$({{m}_{U}})$$  and lower  $$({{m}_{L}})$$  bounds on the mean life (ML estimate of the mean life) are estimated by:
 * $${{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}$$


 * $${{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}$$

where $${{K}_{\alpha }}$$  is defined by:


 * $$\alpha =\frac{1}{\sqrt{2\pi }}\int_^{\infty }{{e}^{-\tfrac{2}}}dt=1-\Phi ({{K}_{\alpha }})$$

If $$\delta $$  is the confidence level, then  $$\alpha =\tfrac{1-\delta }{2}$$  for the two-sided bounds, and  $$\alpha =1-\delta $$  for the one-sided bounds. The variance of $$\widehat{m}$$  is given by:


 * $$\begin{align}

& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial K} \right)}^{2}}Var(\widehat{K})+{{\left( \frac{\partial m}{\partial n} \right)}^{2}}Var(\widehat{n}) +2\left( \frac{\partial m}{\partial K} \right)\left( \frac{\partial m}{\partial n} \right)Cov(\widehat{K},\widehat{n}) \end{align}$$ or:
 * $$Var(\widehat{m})=\frac{1}\left[ \frac{1}Var(\widehat{K})+{{\left[ \ln (V) \right]}^{2}}Var(\widehat{n})+\frac{2\ln (V)}{\widehat{K}}Cov(\widehat{K},\widehat{n}) \right]$$

The variances and covariance of $$K$$  and  $$n$$  are estimated from the Fisher matrix (evaluated at  $$\widehat{K},$$   $$\widehat{n})$$  as follows:
 * $$\left[ \begin{matrix}

Var(\widehat{K}) & Cov(\widehat{K},\widehat{n}) \\ Cov(\widehat{n},\widehat{K}) & Var(\widehat{n}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{K}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial K\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial K} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{n}^{2}}} \\ \end{matrix} \right]}^{-1}}$$

Confidence Bounds on Reliability
The bounds on reliability at a given time, $$T$$, are estimated by:
 * $$\begin{align}

& {{R}_{U}}= & {{e}^{-\tfrac{T}}} \\ & &  \\  & {{R}_{L}}= & {{e}^{-\tfrac{T}}} \end{align}$$ where $${{m}_{U}}$$  and  $${{m}_{L}}$$  are estimated using Eqns. (IPLxpMeanUpper) and (IPLxpMeanLower).

Confidence Bounds on Time
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:


 * $$\widehat{T}=-\widehat{m}\cdot \ln (R)$$

The corresponding confidence bounds are estimated from:


 * $$\begin{align}

& {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align}$$

where $${{m}_{U}}$$  and  $${{m}_{L}}$$  are estimated using Eqns. (IPLxpMeanUpper) and (IPLxpMeanLower).

Bounds on the Parameters
Using the same approach as previously discussed ( $$\widehat{\beta }$$ and  $$\widehat{K}$$  positive parameters):


 * $$\begin{align}

& {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align}$$


 * $$\begin{align}

& {{K}_{U}}= & \widehat{K}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{K})}}{\widehat{K}}}} \\ & {{K}_{L}}= & \widehat{K}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{K})}}{\widehat{K}}}} \end{align}$$


 * and:


 * $$\begin{align}

& {{n}_{U}}= & \widehat{n}+{{K}_{\alpha }}\sqrt{Var(\widehat{n})} \\ & {{n}_{L}}= & \widehat{n}-{{K}_{\alpha }}\sqrt{Var(\widehat{n})} \end{align}$$ The variances and covariances of $$\beta ,$$   $$K,$$  and  $$n$$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{\beta },$$   $$\widehat{K},$$   $$\widehat{n})$$  as follows:
 * $$\left[ \begin{matrix}

Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{K}) & Cov(\widehat{\beta },\widehat{n}) \\ Cov(\widehat{K},\widehat{\beta }) & Var(\widehat{K}) & Cov(\widehat{K},\widehat{n}) \\ Cov(\widehat{n},\widehat{\beta }) & Cov(\widehat{n},\widehat{K}) & Var(\widehat{n}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial K} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial K\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{K}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial K\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{n}^{2}}} \\ \end{matrix} \right]}^{-1}}$$

Confidence Bounds on Reliability
The reliability function (ML estimate) for the IPL-Weibull model is given by:


 * $$\widehat{R}(T,V)={{e}^{-{{\left( \widehat{K}{{V}^{\widehat{n}}}T \right)}^{\widehat{\beta }}}}}$$


 * or:


 * $$\widehat{R}(T,V)={{e}^{-{{e}^{\ln \left[ {{\left( \widehat{K}{{V}^{\widehat{n}}}T \right)}^{\widehat{\beta }}} \right]}}}}$$


 * Setting:


 * $$\widehat{u}=\ln \left[ \left( \widehat{K}V\widehat{^{n}}T \right)\widehat{^{\beta }} \right]$$


 * or:


 * $$\widehat{u}=\widehat{\beta }\left[ \ln (T)+\ln (\widehat{K})+\widehat{n}\ln (V) \right]$$

The reliability function now becomes:


 * $$\widehat{R}(T,V)={{e}^{-e\widehat{^{u}}}}$$

The next step is to find the upper and lower bounds on $$\widehat{u}$$ :


 * $${{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}$$


 * $${{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}$$


 * where:


 * $$\begin{align}

& Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial K} \right)}^{2}}Var(\widehat{K}) \\ & & +{{\left( \frac{\partial \widehat{u}}{\partial n} \right)}^{2}}Var(\widehat{n}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial K} \right)Cov(\widehat{\beta },\widehat{K})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial n} \right)Cov(\widehat{\beta },\widehat{n}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial K} \right)\left( \frac{\partial \widehat{u}}{\partial n} \right)Cov(\widehat{K},\widehat{n}) \end{align}$$


 * or:


 * $$\begin{align}

& Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{\widehat{K}} \right)}^{2}}Var(\widehat{K}) \\ & & +{{\widehat{\beta }}^{2}}{{\left[ \ln (V) \right]}^{2}}Var(\widehat{n}) \\ & & +\frac{2\widehat{u}}{\widehat{K}}Cov(\widehat{\beta },\widehat{K})+2\widehat{u}\ln (V)Cov(\widehat{\beta },\widehat{n})+\frac{2{{\widehat{\beta }}^{2}}\ln (V)}{\widehat{K}}Cov(\widehat{K},\widehat{n}) \end{align}$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align}$$

where $${{u}_{U}}$$  and  $${{u}_{L}}$$  are estimated using Eqns. (IPLUupper) and (IPLUlower).

Confidence Bounds on Time
The bounds on time for a given reliability (ML estimate of time) are estimated by first solving the reliability function with respect to time:


 * $$\begin{align}

& \ln (R)= & -{{\left( \widehat{K}{{V}^{\widehat{n}}}\widehat{T} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left[ \ln (\widehat{T})+\ln (\widehat{K})+\widehat{n}\ln (V) \right] \end{align}$$


 * or:


 * $$\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln (\widehat{K})-\widehat{n}\ln (V)$$

where $$\widehat{u}=\ln \widehat{T}.$$  The upper and lower bounds on  $$u$$  are estimated from:


 * $$\begin{align}

& {{u}_{U}}= & \widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \\ & {{u}_{L}}= & \widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \end{align}$$


 * where:


 * $$\begin{align}

& Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial K} \right)}^{2}}Var(\widehat{K}) \\ & & +{{\left( \frac{\partial \widehat{u}}{\partial n} \right)}^{2}}Var(\widehat{n}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial K} \right)Cov(\widehat{\beta },\widehat{K}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial n} \right)Cov(\widehat{\beta },\widehat{n}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial K} \right)\left( \frac{\partial \widehat{u}}{\partial n} \right)Cov(\widehat{K},\widehat{n}) \end{align}$$


 * or:


 * $$\begin{align}

& Var(\widehat{u})= & \frac{1}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}Var(\widehat{K}) \\ & & +{{\left[ \ln (V) \right]}^{2}}Var(\widehat{n}) \\ & & +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{K}}Cov(\widehat{\beta },\widehat{K}) \\ & & +\frac{2\ln (-\ln (R))}\ln (V)Cov(\widehat{\beta },\widehat{n}) \\ & & +\frac{2\ln (V)}{\widehat{K}}Cov(\widehat{K},\widehat{n}) \end{align}$$

The upper and lower bounds on time are then found by:


 * $$\begin{align}

& {{T}_{U}}= & {{e}^} \\ & {{T}_{L}}= & {{e}^} \end{align}$$

where $${{u}_{U}}$$  and  $${{u}_{L}}$$  are estimated using Eqns. (IPLeibTupper) and (IPLeibTlower).

Bounds on the Parameters
Since the standard deviation, $${{\widehat{\sigma }}_{T}}$$, and  $$\widehat{K}$$  are positive parameters, then  $$\ln ({{\widehat{\sigma }}_})$$  and  $$\ln (\widehat{K})$$  are treated as normally distributed, and the bounds are estimated from:
 * $$\begin{align}

& {{\sigma }_{U}}= & {{\widehat{\sigma }}_}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_})}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac\text{ (Lower bound)} \end{align}$$


 * and:


 * $$\begin{align}

& {{K}_{U}}= & \widehat{K}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{K})}}{\widehat{K}}}}\text{ (Upper bound)} \\ & {{K}_{L}}= & \frac{\widehat{K}}\text{ (Lower bound)} \end{align}$$

The lower and upper bounds on $$n$$, are estimated from:


 * $$\begin{align}

& {{n}_{U}}= & \widehat{n}+{{K}_{\alpha }}\sqrt{Var(\widehat{n})}\text{ (Upper bound)} \\ & {{n}_{L}}= & \widehat{n}-{{K}_{\alpha }}\sqrt{Var(\widehat{n})}\text{ (Lower bound)} \end{align}$$

The variances and covariances of $$A,$$   $$B,$$  and  $${{\sigma }_}$$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{A},$$   $$\widehat{B}$$,  $${{\widehat{\sigma }}_}),$$  as follows:


 * $$\left[ \begin{matrix}

Var({{\widehat{\sigma }}_}) & Cov(\widehat{K},{{\widehat{\sigma }}_}) & Cov(\widehat{n},{{\widehat{\sigma }}_}) \\ Cov({{\widehat{\sigma }}_},\widehat{K}) & Var(\widehat{K}) & Cov(\widehat{K},\widehat{n}) \\ Cov({{\widehat{\sigma }}_},\widehat{n}) & Cov(\widehat{n},\widehat{K}) & Var\left( \widehat{n} \right) \\ \end{matrix} \right]={{\left[ F \right]}^{-1}}$$


 * where:


 * $$F=\left[ \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial K} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial K\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{K}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial K\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial K} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{n}^{2}}} \\ \end{matrix} \right]$$

Bounds on Reliability
The reliability of the lognormal distribution is:


 * $$R({T}',V;K,n,{{\sigma }_})=\mathop{}_^{\infty }\frac{1}{{{\widehat{\sigma }}_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (\widehat{K})+\widehat{n}\ln (V)} \right)}^{2}}}}dt$$

Let $$\widehat{z}(t,V;K,n,{{\sigma }_{T}})=\tfrac{t+\ln (\widehat{K})+\widehat{n}\ln (V)},$$  then  $$\tfrac{d\widehat{z}}{dt}=\tfrac{1}.$$

For $$t={T}'$$,  $$\widehat{z}=\tfrac{{T}'+\ln (\widehat{K})+\widehat{n}\ln (V)}$$ , and for  $$t=\infty ,$$   $$\widehat{z}=\infty .$$  The above equation then becomes:


 * $$R(\widehat{z})=\mathop{}_{\widehat{z}({T}',V)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The bounds on $$z$$  are estimated from:


 * $$\begin{align}

& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align}$$


 * where:

.
 * $$\begin{align}

& Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial K} \right)_{\widehat{K}}^{2}Var(\widehat{K})+\left( \frac{\partial \widehat{z}}{\partial n} \right)_{\widehat{n}}^{2}Var(\widehat{n})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)_^{2}Var({{\widehat{\sigma }}_{T}}) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial K} \right)}_{\widehat{K}}}{{\left( \frac{\partial \widehat{z}}{\partial n} \right)}_{\widehat{n}}}Cov\left( \widehat{K},\widehat{n} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial K} \right)}_{\widehat{K}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)}_}Cov\left( \widehat{K},{{\widehat{\sigma }}_{T}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial n} \right)}_{\widehat{n}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)}_}Cov\left( \widehat{n},{{\widehat{\sigma }}_{T}} \right) \end{align}$$. or:


 * $$\begin{align}

& Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_^{2}}[\frac{1}Var(\widehat{K})+\ln {{(V)}^{2}}Var(\widehat{n})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_}) \\ & & +\frac{2\ln (V)}{K}Cov\left( \widehat{K},\widehat{n} \right)-\frac{2\widehat{z}}{K}Cov\left( \widehat{K},{{\widehat{\sigma }}_} \right)-2\widehat{z}\ln (V)Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right)] \end{align}$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & \mathop{}_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align}$$

Confidence Bounds on Time
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:


 * $${T}'(V;\widehat{K},\widehat{n},{{\widehat{\sigma }}_})=-\ln (\widehat{K})-\widehat{n}\ln (V)+z\cdot {{\widehat{\sigma }}_}$$

where:


 * $$\begin{align}

& {T}'(V;\widehat{K},\widehat{n},{{\widehat{\sigma }}_})= & \ln (T) \\ & z= & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align}$$

and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The next step is to calculate the variance of $${T}'(V;\widehat{K},\widehat{n},{{\widehat{\sigma }}_}):$$


 * $$\begin{align}

& Var({T}')= & {{\left( \frac{\partial {T}'}{\partial K} \right)}^{2}}Var(\widehat{K})+{{\left( \frac{\partial {T}'}{\partial n} \right)}^{2}}Var(\widehat{n})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)}^{2}}Var({{\widehat{\sigma }}_}) \\ & & +2\left( \frac{\partial {T}'}{\partial K} \right)\left( \frac{\partial {T}'}{\partial n} \right)Cov\left( \widehat{K},\widehat{n} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial K} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)Cov\left( \widehat{K},{{\widehat{\sigma }}_} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial n} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right) \end{align}$$

or:


 * $$\begin{align}

& Var({T}')= & \frac{1}Var(\widehat{K})+\ln {{(V)}^{2}}Var(\widehat{n})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_}) \\ & & +\frac{2\ln (V)}{K}Cov\left( \widehat{K},\widehat{n} \right) \\ & & -\frac{2\widehat{z}}{K}Cov\left( \widehat{K},{{\widehat{\sigma }}_} \right) \\ & & -2\widehat{z}\ln (V)Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right) \end{align}$$

The upper and lower bounds are then found by:


 * $$\begin{align}

& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align}$$

Solving for $${{T}_{U}}$$  and  $${{T}_{L}}$$  yields:


 * $$\begin{align}

& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align}$$