Template:Bounds on reliability rsa

Fisher Matrix Bounds
These bounds are based on:


 * $$\log it(\widehat{R}(t))\sim N(0,1)$$


 * $$\log it(\widehat{R}(t))=\ln \left\{ \frac{\widehat{R}(t)}{1-\widehat{R}(t)} \right\}$$

The confidence bounds on reliability are given by:


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


 * $$Var(\widehat{R}(t))={{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+2\left( \frac{\partial R}{\partial \beta } \right)\left( \frac{\partial R}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })$$

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


 * $$\begin{align}

& \frac{\partial R}{\partial \beta }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[\lambda {{t}^{\widehat{\beta }}}\ln (t)-\lambda {{(t+d)}^{\widehat{\beta }}}\ln (t+d)] \\ & \frac{\partial R}{\partial \lambda }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[{{t}^{\widehat{\beta }}}-{{(t+d)}^{\widehat{\beta }}}] \end{align}$$

Crow Bounds
Failure Terminated Data With failure terminated data, the 100( $$1-\alpha $$ )% confidence interval for the current reliability at time $$t$$  in a specified mission time  $$d$$  is:


 * $$({{[\widehat{R}(d)]}^{\tfrac{1}}},{{[\hat{R}(d)]}^{\tfrac{1}}})$$


 * where


 * $$\widehat{R}(\tau )={{e}^{-[\widehat{\lambda }{{(t+\tau )}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}$$

$${{p}_{1}}$$ and $${{p}_{2}}$$  can be obtained from Eqn. (ft). Time Terminated Data With time terminated data, the 100( $$1-\alpha $$ )% confidence interval for the current reliability at time $$t$$  in a specified mission time  $$\tau $$  is:


 * $$({{[\widehat{R}(d)]}^{\tfrac{1}}},{{[\hat{R}(d)]}^{\tfrac{1}}})$$


 * where:


 * $$\widehat{R}(d)={{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}$$

$${{p}_{1}}$$ and  $${{p}_{2}}$$  can be obtained from Eqn. (tt).