Template:Bounds on growth potential mtbf rga

Bounds on Growth Potential MTBF

Fisher Matrix Bounds

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

where [math]\displaystyle{ {{r}_{U}} }[/math] and [math]\displaystyle{ {{r}_{L}} }[/math] can be obtained from Eqn. (GPR).

Crow Bounds

The Crow bounds for the growth potential MTBF are the same as the Fisher Matrix bounds.

Example 3
Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the data in Table 9.1.

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 . 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]

Figure extendedpic7 shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity. Figure extendedpic8 shows these bounds based on the Crow method.

Fisher Matrix confidence bounds for the failure intensity.

Crow confidence bounds for the failure intensity.

  Example 4
Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the data in Table 9.3.

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 }}}_{i}}(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 }}}_{i}}(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{ \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 }[/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]. [math]\displaystyle{ \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 }[/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]

Figure extendedpic9 shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF. Figure extendedpic10 shows these bounds based on the Crow method.

Fisher Matrix confidence bounds on MTBF.

Crow confidence bounds on MTBF.