Template:Bounds on time given instantaneous failure intensity rsa

Bounds on Time Given Instantaneous Failure Intensity

Fisher Matrix Bounds

These bounds are based on:

[math]\displaystyle{ \frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\widehat{T}) \right]}}\sim N(0,1) }[/math]

The confidence bounds on the time are given by:

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

where:

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

The variance calculation is the same as Eqns. (var1), (var2) and (var3).

[math]\displaystyle{ \widehat{T}={{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}} }[/math]

[math]\displaystyle{ \begin{align} & \frac{\partial T}{\partial \beta }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}[-\frac{\ln (\tfrac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta })}{{{(\beta -1)}^{2}}}+\frac{1}{\beta (1-\beta )}] \\ & \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} \end{align} }[/math]

Crow Bounds

Step 1: Calculate [math]\displaystyle{ {{\lambda }_{i}}(T)=\tfrac{1}{MTB{{F}_{i}}} }[/math] .
Step 2: Use the equations from 13.1.7.9 to calculate the bounds on time given the instantaneous failure intensity.