Power Law Model Confidence Bounds Example

These examples appear in the Reliability growth reference.

Using the data from the Power Law Model Parameter Estimation Example power law model example given above, 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 the example are:


 * $$\begin{align}

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

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}\,\!$$

The next two figures show the Fisher matrix and Crow confidence bounds on mission reliability for mission time $$d=40\,\!$$.