Crow Extended Confidence Bounds Example

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This example appears in the Reliability growth reference.

Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the Test-Find-Test data shown in the Crow Extended Test-Find-Test Data example.

Solution

The estimated demonstrated failure intensity is [math]\displaystyle{ {{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\! }[/math]. Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:

[math]\displaystyle{ \begin{align} {{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.08152 \end{align}\,\! }[/math]

[math]\displaystyle{ \begin{align} {{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.13525 \end{align}\,\! }[/math]

The Crow confidence bounds for the demonstrated failure intensity at the 90% confidence level are:

[math]\displaystyle{ \begin{align} {{[{{\lambda }_{D}}(T)]}_{L}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\ = & 0.07985 \\ {{[{{\lambda }_{D}}(T)]}_{U}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N} \\ = & 0.13299 \end{align}\,\! }[/math]

The projected failure intensity is:

[math]\displaystyle{ \begin{align} \hat{\lambda_{p}} &= \frac{N_{i}}{T}+\sum_{i=1}^{M}(1-d_{i})\frac{N}{T}+\overline{d}\left(\frac{M}{T}\overline{\beta} \right )\\ &= 0.06611 \end{align} }[/math]

Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:

[math]\displaystyle{ \begin{align} {{[{{{\hat{\lambda }}}_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.04902 \end{align}\,\! }[/math]

[math]\displaystyle{ \begin{align} {{[{{{\hat{\lambda }}}_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.08915 \end{align}\,\! }[/math]

The Crow confidence bounds for the projected failure intensity are:

[math]\displaystyle{ \begin{align} {{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.04807 \\ {{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.09090 \end{align}\,\! }[/math]

The growth potential failure intensity is:

[math]\displaystyle{ \widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\! }[/math].

Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:

[math]\displaystyle{ \begin{align} {{r}_{L}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.03020 \\ {{r}_{U}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.0656 \end{align}\,\! }[/math]

The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity.

The following figure shows these bounds based on the Crow method.