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==Confidence Bounds for Competing Failure Modes==

The method available in Weibull++ for estimating the different types of confidence bounds, for competing failure modes analysis, is the Fisher matrix method, and is presented in this section.

===Variance/Covariance Matrix===

The variances and covariances of the parameters are estimated from the inverse local Fisher matrix, as follows:

<math>\begin{align}
  & \left( \begin{matrix}
   Var({{{\hat{a}}}_{1}}) & Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0  \\
   Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & Var({{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0  \\
   0 & 0 & \cdot  & 0 & 0 & 0 & 0  \\
   0 & 0 & 0 & \cdot  & 0 & 0 & 0  \\
   0 & 0 & 0 & 0 & \cdot  & 0 & 0  \\
   0 & 0 & 0 & 0 & 0 & Var({{{\hat{a}}}_{n}}) & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}})  \\
   0 & 0 & 0 & 0 & 0 & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) & Var({{{\hat{b}}}_{n}})  \\
\end{matrix} \right) \\ 
 & =\left( \begin{matrix}
   -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & 0 & 0 & 0 & 0 & 0  \\
   -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{1}^{2}} & 0 & 0 & 0 & 0 & 0  \\
   0 & 0 & \cdot  & 0 & 0 & 0 & 0  \\
   0 & 0 & 0 & \cdot  & 0 & 0 & 0  \\
   0 & 0 & 0 & 0 & \cdot  & 0 & 0  \\
   0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}}  \\
   0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{n}^{2}}  \\
\end{matrix} \right) \\ 
\end{align}</math>

where <math>\Lambda </math> is the log-likelihood function of the failure distribution, described in Chapter [[Parameter Estimation]].

===Bounds on Reliability===

The competing failure modes reliability function is given by: 

::<math>\widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}}</math>

where:
::•	 <math>{{R}_{i}}</math>  is the reliability of the  <math>{{i}^{th}}</math>  mode,
::•	 <math>n</math>  is the number of failure modes.

The upper and lower bounds on reliability are estimated using the logit transformation: 

::<math>\begin{align}
  & {{R}_{U}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}} \\ 
 & {{R}_{L}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}}  
\end{align}</math>

where  <math>\widehat{R}</math>  is calculated using the reliability equation for competing failure modes.
<math>{{K}_{\alpha }}</math>  is defined by: 

::<math>\alpha =\frac{1}{\sqrt{2\pi }}\underset{{{K}_{\alpha }}}{\overset{\infty }{\mathop \int }}\,{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>

(If  <math>\delta </math>  is the confidence level, then  <math>\alpha =\tfrac{1-\delta }{2}</math>  for the two-sided bounds, and  <math>\alpha =1-\delta </math>  for the one-sided bounds.)

The variance of  <math>\widehat{R}</math>  is estimated by: 

::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial R}{\partial {{R}_{i}}} \right)}^{2}}Var({{\hat{R}}_{i}})</math>

::<math>\frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{{{R}_{j}}}</math>

Thus: 

::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( \underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{R}_{j}^{2} \right)Var({{\hat{R}}_{i}})</math>

::<math>Var({{\hat{R}}_{i}})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial {{R}_{i}}}{\partial {{a}_{i}}} \right)}^{2}}Var({{\hat{a}}_{i}})</math>

where  <math>\widehat{{{a}_{i}}}</math>  is an element of the model parameter vector.  

Therefore, the value of  <math>Var({{\hat{R}}_{i}})</math>  is dependent on the underlying distribution.


For the Weibull distribution: 

::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}})</math>

where: 

::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})</math>

and  <math>Var(\widehat{{{u}_{i}}})</math>  is given in Chapter [[The Weibull Distribution]].


For the exponential distribution: 

::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})</math>

where  <math>Var(\widehat{{{\lambda }_{i}}})</math>  is given in Chapter [[The Exponential Distribution]].


For the normal distribution: 

::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\hat{\sigma } \right)}^{2}}Var({{\hat{z}}_{i}})</math>

::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}</math>

where  <math>Var(\widehat{{{z}_{i}}})</math>  is given in Chapter [[The Normal Distribution]].


For the lognormal distribution: 

::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}})</math>

::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}</math>

where  <math>Var(\widehat{{{z}_{i}}})</math>  is given in Chapter [[The Lognormal Distribution]].

===Bounds on Time===
The bounds on time are estimate by solving the reliability equation with respect to time. From the reliabilty equation for competing faiure modes, we have that: 

::<math>\hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}})</math>
 
::<math>i=1,...,n</math>

where:
:•	 <math>\varphi </math>  is inverse function for the reliabilty equation for competing faiure modes.
:•	for the Weibull distribution  <math>{{\hat{a}}_{i}}</math>  is  <math>{{\hat{\beta }}_{i}}</math> , and  <math>{{\hat{b}}_{i}}</math>  is  <math>{{\hat{\eta }}_{i}}</math> 
:•	for the exponential distribution  <math>{{\hat{a}}_{i}}</math>  is  <math>{{\hat{\lambda }}_{i}}</math> , and  <math>{{\hat{b}}_{i}}</math>  =0
:•	for the normal distribution  <math>{{\hat{a}}_{i}}</math>  is  <math>{{\hat{\mu }}_{i}}</math> , and  <math>{{\hat{b}}_{i}}</math>  is  <math>{{\hat{\sigma }}_{i}}</math> , and
:•	for the lognormal distribution  <math>{{\hat{a}}_{i}}</math>  is  <math>\hat{\mu }_{i}^{\prime }</math> , and  <math>{{\hat{b}}_{i}}</math>  is  <math>\hat{\sigma }_{i}^{\prime }</math> 

Set: 

::<math>u=\ln (t)</math>

The bounds on  <math>u</math>  are estimated from: 

::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

and: 

::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

Then the upper and lower bounds on time are found by using the equations 

::<math>{{t}_{U}}={{e}^{{{u}_{U}}}}</math>

and: 

::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}</math>

<math>{{K}_{\alpha }}</math>   is calculated using the inverse standard normal distribution and  <math>Var(\widehat{u})</math>  is computed as: 

::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)</math>