Template:Bounds on demonstrated failure intensity rga

Bounds on Demonstrated Failure Intensity

Fisher Matrix Bounds

If there are no BC failure modes, the demonstrated failure intensity is [math]\displaystyle{ {{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T} }[/math] . Thus:

[math]\displaystyle{ Var({{\hat{\lambda }}_{D}}(t))=\frac{{{N}_{A}}}{{{T}^{2}}}+\frac{{{N}_{BD}}}{{{T}^{2}}}=\frac{{{\lambda }_{D}}(t)}{T} }[/math]

and:

[math]\displaystyle{ \sqrt{T}\left( \frac{{{{\hat{\lambda }}}_{D}}(T)-{{\lambda }_{D}}(T)}{\sqrt{{{\lambda }_{D}}(T)}} \right)\sim N(0,1) }[/math]

[math]\displaystyle{ {{\lambda }_{D}}(T)={{\hat{\lambda }}_{D}}(T)+\frac{{{C}^{2}}}{2}\pm \sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} }[/math]

where [math]\displaystyle{ C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}} }[/math] .
If there are BC failure modes, the demonstrated failure intensity, [math]\displaystyle{ {{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}} }[/math] , is actually the instantaneous failure intensity based on all of the data. [math]\displaystyle{ {{\lambda }_{CA}}(T) }[/math] must be positive, thus [math]\displaystyle{ \ln {{\lambda }_{CA}}(T) }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim N(0,1) }[/math]

The approximate confidence bounds on the instantaneous failure intensity are then estimated from:

[math]\displaystyle{ CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} }[/math]

where [math]\displaystyle{ {{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}} }[/math] .

[math]\displaystyle{ \begin{align} & Var({{{\hat{\lambda }}}_{CA}}(T))= & {{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & & +2\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) \end{align} }[/math]

The variance calculation is the same as described in Chapter 5.

Crow Bounds

If there are no BC failure modes then:

[math]\displaystyle{ \begin{align} & {{[{{\lambda }_{D}}(T)]}_{l}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\ & {{[{{\lambda }_{D}}(T)]}_{u}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N} \end{align} }[/math]

where [math]\displaystyle{ {{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}} }[/math] .
If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in Chapter 5.