|
|
| (One intermediate revision by the same user not shown) |
| Line 1: |
Line 1: |
| ===Bounds on Growth Potential MTBF===
| | #REDIRECT [[Crow Extended Confidence Bounds]] |
| ====Fisher Matrix Bounds====
| |
| ::<math>\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>{{r}_{U}}</math> and <math>{{r}_{L}}</math> can be obtained from Eqn. (GPR).
| |
| <br>
| |
| ====Crow Bounds====
| |
| The Crow bounds for the growth potential MTBF are the same as the Fisher Matrix bounds.
| |
| <br>
| |
| <br>
| |
| '''Example 3'''
| |
| <br>
| |
| Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the data in Table 9.1.
| |
| <br>
| |
| <br>
| |
| '''Solution'''
| |
| <br>
| |
| The estimated demonstrated failure intensity is <math>{{\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>\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>\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>\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>\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>\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>\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>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 </math>.
| |
| | |
| <br>
| |
| Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:
| |
| | |
| ::<math>\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.
| |
| <br>
| |
| <br>
| |
| [[Image:rga9.8.png|thumb|center|400px|Fisher Matrix confidence bounds for the failure intensity.]]
| |
| <br>
| |
| <br>
| |
| [[Image:rga9.9.png|thumb|center|400px|Crow confidence bounds for the failure intensity.]]
| |
| | |
|
| |
| '''Example 4'''
| |
| <br>
| |
| 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.
| |
| <br>
| |
| <br>
| |
| '''Solution'''
| |
| <br>
| |
| For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, <math>{{\hat{\lambda }}_{D}}(T)</math> , is simply the Crow-AMSAA model applied to all A, BC, and BD data.
| |
| | |
| ::<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744</math>
| |
| | |
| Therefore, the demonstrated MTBF is:
| |
| | |
| ::<math>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>\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>\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>\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>\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>
| |
| | |
| <br>
| |
| The projected failure intensity is <math>\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>\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>\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>\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>\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>\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>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 </math>.
| |
| <math>\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>\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>\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.
| |
|
| |
| [[Image:rga9.10.png|thumb|center|400px|Fisher Matrix confidence bounds on MTBF.]]
| |
| <br>
| |
| <br>
| |
| [[Image:rga9.11.png|thumb|center|400px|Crow confidence bounds on MTBF.]]
| |