Template:Bounds on cumulative number of failures rsa

Fisher Matrix Bounds
The cumulative number of failures, $$N(t)$$, must be positive, thus  $$\ln \left( N(t) \right)$$  is approximately treated as being normally distributed.


 * $$\frac{\ln (\widehat{N}(t))-\ln (N(t))}{\sqrt{Var\left[ \ln \widehat{N}(t) \right]}}\sim N(0,1)$$


 * $$N(t)=\widehat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{N}(t))}/\widehat{N}(t)}}$$


 * where:


 * $$\widehat{N}(t)=\widehat{\lambda }{{t}^{\widehat{\beta }}}$$


 * $$\begin{align}

& Var(\widehat{N}(t))= & {{\left( \frac{\partial N(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial N(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\ & & +2\left( \frac{\partial N(t)}{\partial \beta } \right)\left( \frac{\partial N(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda }) \end{align}$$

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


 * $$\begin{align}

& \frac{\partial N(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\widehat{\beta }}}\ln (t) \\ & \frac{\partial N(t)}{\partial \lambda }= & t\widehat{\beta } \end{align}$$

Crow Bounds

 * $$\begin{array}{*{35}{l}}

{{N}_{L}}(T)=\tfrac{T}{\widehat{\beta }}{{\lambda }_{i}}{{(T)}_{L}} \\ {{N}_{U}}(T)=\tfrac{T}{\widehat{\beta }}{{\lambda }_{i}}{{(T)}_{U}} \\ \end{array}$$

where $${{\lambda }_{i}}{{(T)}_{L}}$$  and  $${{\lambda }_{i}}{{(T)}_{U}}$$  can be obtained from Eqn. (inr).

Example 3
Using the data from Example 1, calculate the mission reliability at $$t=2000$$  hours and mission time  $$d=40$$  hours  along with the confidence bounds at the 90% confidence level. Solution The maximum likelihood estimates of $$\widehat{\lambda }$$  and  $$\widehat{\beta }$$  from Example 1 are:


 * $$\begin{align}

& \widehat{\beta }= & 0.45300 \\ & \widehat{\lambda }= & 0.36224 \end{align}$$

From Eq. (reliability), the mission reliability at $$t=2000$$  for mission time  $$d=40$$  is:


 * $$\begin{align}

& \widehat{R}(t)= & {{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} \\ & = & 0.90292 \end{align}$$

At the 90% confidence level and $$T=2000$$  hours, the Fisher Matrix confidence bounds for the mission reliability for mission time  $$d=40$$  are given by:


 * $$CB=\frac{\widehat{R}(t)}{\widehat{R}(t)+(1-\widehat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{R}(t))}/\left[ \widehat{R}(t)(1-\widehat{R}(t)) \right]}}}$$


 * $$\begin{align}

& {{[\widehat{R}(t)]}_{L}}= & 0.83711 \\ & {{[\widehat{R}(t)]}_{U}}= & 0.94392 \end{align}$$

The Crow confidence bounds for the mission reliability are:


 * $$\begin{align}

& {{[\widehat{R}(t)]}_{L}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}}} \\ & = & {{[0.90292]}^{\tfrac{1}{0.71440}}} \\ & = & 0.86680 \\ & {{[\widehat{R}(t)]}_{U}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}}} \\ & = & {{[0.90292]}^{\tfrac{1}{1.6051}}} \\ & = & 0.93836 \end{align}$$

Figures ConfReliFish and ConfRelCrow show the Fisher Matrix and Crow confidence bounds on mission reliability for mission time $$d=40$$.