Crow Extended Confidence Bounds Example 2

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This example appears in the Reliability growth reference.

Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the Test-Fix-Find-Test data in the Crow Extended Test-Fix-Find-Test Example.

Solution
For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, [math]\displaystyle{ {{\hat{\lambda }}_{D}}(T)\,\! }[/math], is simply the Crow-AMSAA model applied to all A, BC, and BD data.

[math]\displaystyle{ {{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\! }[/math]

Therefore, the demonstrated MTBF is:

[math]\displaystyle{ MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\! }[/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 }}}_{CA}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{CA}}(T)}} \\ = & 0.09339 \end{align}\,\! }[/math]

[math]\displaystyle{ \begin{align} {{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{CA}}(T)}} \\ = & 0.17390 \end{align}\,\! }[/math]

The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:

[math]\displaystyle{ \begin{align} MTB{{F}_{{{D}_{L}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{U}}} \\ = & 5.75054 \\ MTB{{F}_{{{D}_{U}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{L}}} \\ = & 10.70799 \end{align}\,\! }[/math]

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

[math]\displaystyle{ \begin{align} MTB{{F}_{{{D}_{L}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{U}}} \\ = & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,\alpha /2)}{2N}} \\ = & 5.6325 \\ MTB{{F}_{{{D}_{U}}}} = & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{L}}} \\ = & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,1-\alpha /2)}{2N}} \\ = & 10.8779 \end{align}\,\! }[/math]

The projected failure intensity is:

[math]\displaystyle{ \begin{align} \hat{\lambda}_P (T) &= \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} + \bar{d}\widehat{h}(T|BD) \\ &= 0.0885 \,\! \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} {{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.0681 \end{align}\,\! }[/math]

[math]\displaystyle{ \begin{align} {{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.1152 \end{align}\,\! }[/math]

The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:

[math]\displaystyle{ \begin{align} MTB{{F}_{{{P}_{L}}}} = & \frac{1}{{{[{{\lambda }_{P}}(T)]}_{U}}} \\ = & 8.6818 \\ MTB{{F}_{{{P}_{U}}}} = & \frac{1}{{{[{{\lambda }_{P}}(T)]}_{L}}} \\ = & 14.6926 \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}} \\ = & 0.0672 \\ {{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.1166 \end{align}\,\! }[/math]

The Crow confidence bounds for the projected MTBF at the 90% confidence level are:

[math]\displaystyle{ \begin{align} MTB{{F}_{{{P}_{L}}}} = & \frac{1}{{{[{{\widehat{\lambda }}_{P}}(T)]}_{U}}} \\ = & 8.5743 \\ MTB{{F}_{{{P}_{U}}}} = & \frac{1}{{{[{{\widehat{\lambda }}_{P}}(T)]}_{L}}} \\ = & 14.8769 \end{align}\,\! }[/math]

The growth potential failure intensity is:

[math]\displaystyle{ \widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\! }[/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}} \\ = & 0.0488 \\ {{r}_{U}} = & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ = & 0.0919 \end{align}\,\! }[/math]

The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:

[math]\displaystyle{ \begin{align} MTB{{F}_{G{{P}_{L}}}} = & \frac{1}{{{r}_{U}}} \\ = & 10.8790 \\ MTB{{F}_{G{{P}_{U}}}} = & \frac{1}{{{r}_{L}}} \\ = & 20.4855 \end{align}\,\! }[/math]

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

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