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| ===Bounds on Cumulative Number of Failures===
| | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_Number_of_Failures]] |
| ====Fisher Matrix Bounds====
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| The cumulative number of failures, <math>N(t)</math> , must be positive, thus <math>\ln N(t)</math> is treated as being normally distributed.
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| ::<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)</math>
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| ::<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}</math>
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| <br>
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| :where:
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| ::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
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| ::<math>\begin{align}
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| & Var(\hat{N}(t))= & {{\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
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| & & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })
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| \end{align}</math>
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| The variance calculation is the same as Eqn. (variance1) and:
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| ::<math>\begin{align}
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| & \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\
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| & \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}
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| \end{align}</math>
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| ====Crow Bounds====
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| <br>
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| The Crow cumulative number of failure confidence bounds are:
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| ::<math>\begin{align}
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| & {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\
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| & {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}}
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| \end{align}</math>
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| <br>
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| where <math>{{\lambda }_{i}}{{(T)}_{L}}</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}</math> can be obtained from Eqn. (amsaac14).
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| <br>
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| <br>
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| '''Example 2'''
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| <br>
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| Calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity for the data from Example 1 given in Table 5.1.
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| '''Solution'''
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| '''Fisher Matrix Bounds'''
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| <br>
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| Using <math>\widehat{\beta }</math> and <math>\widehat{\lambda }</math> estimated in Example 1, Eqns. (lambda2partial), (beta2partial) and (lambdabeta2partial) are:
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| ::<math>\begin{align}
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| & \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{22}{{{0.4239}^{2}}}=-122.43 \\
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| & \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}= & -\frac{22}{{{0.6142}^{2}}}-0.4239\cdot {{620}^{0.6142}}{{(\ln 620)}^{2}}=-967.68 \\
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| & \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -{{620}^{0.6142}}\ln 620=-333.64
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| \end{align}</math>
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| <br>
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| The Fisher Matrix then becomes:
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|
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| <br>
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| For <math>T=620</math> hr, the partial derivatives of the cumulative and instantaneous failure intensities are:
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial {{\lambda }_{c}}(T)}{\partial \beta }= & \widehat{\lambda }{{T}^{\widehat{\beta }-1}}\ln (T) \\
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| & = & 0.4239\cdot {{620}^{-0.3858}}\ln 620 \\
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| & = & 0.22811336 \\
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| & \frac{\partial {{\lambda }_{c}}(T)}{\partial \lambda }= & {{T}^{\widehat{\beta }-1}} \\
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| & = & {{620}^{-0.3858}} \\
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| & = & 0.083694185
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| \end{align}</math>
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial {{\lambda }_{i}}(T)}{\partial \beta }= & \widehat{\lambda }{{T}^{\widehat{\beta }-1}}+\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}\ln T \\
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| & = & 0.4239\cdot {{620}^{-0.3858}}+0.4239\cdot 0.6142\cdot {{620}^{-0.3858}}\ln 620 \\
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| & = & 0.17558519
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| \end{align}</math>
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial {{\lambda }_{i}}(T)}{\partial \lambda }= & \widehat{\beta }{{T}^{\widehat{\beta }-1}} \\
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| & = & 0.6142\cdot {{620}^{-0.3858}} \\
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| & = & 0.051404969
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| \end{align}</math>
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| Therefore, the variances become:
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| <br>
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| The cumulative and instantaneous failure intensities at <math>T=620</math> hr are:
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| <br>
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| ::<math>\begin{align}
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| & {{\lambda }_{c}}(T)= & 0.03548 \\
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| & {{\lambda }_{i}}(T)= & 0.02179
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| \end{align}</math>
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| So, at the 90% confidence level and for <math>T=620</math> hr, the Fisher Matrix confidence bounds for the cumulative failure intensity are:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{c}}(T)]}_{L}}= & 0.02499 \\
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| & {{[{{\lambda }_{c}}(T)]}_{U}}= & 0.05039
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| \end{align}</math>
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| The confidence bounds for the instantaneous failure intensity are:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{i}}(T)]}_{L}}= & 0.01327 \\
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| & {{[{{\lambda }_{i}}(T)]}_{U}}= & 0.03579
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| \end{align}</math>
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| Figures 4fig82 and 4fig83 display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.
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| <br>
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| [[Image:rga5.2.png|thumb|center|400px|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]] | |
| <br>
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| [[Image:rga5.3.png|thumb|center|400px|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]
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| <br>
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| '''Crow Bounds'''
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| <br>
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| The Crow confidence bounds for the cumulative failure intensity at the 90% confidence level and for <math>T=620</math> hr are:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{c}}(T)]}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\
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| & = & \frac{29.787476}{2*620} \\
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| & = & 0.02402 \\
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| & {{[{{\lambda }_{c}}(T)]}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t} \\
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| & = & \frac{62.8296}{2*620} \\
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| & = & 0.05067
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| \end{align}</math>
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| The Crow confidence bounds for the instantaneous failure intensity at the 90% confidence level and for <math>T=620</math> hr are:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\
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| & = & \frac{1}{MTB{{F}_{i}}\cdot U} \\
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| & = & 0.01179
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| \end{align}</math>
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}} \\
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| & = & \frac{1}{MTB{{F}_{i}}\cdot L} \\
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| & = & 0.03253
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| \end{align}</math>
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| Figures 4fig84 and 4fig85 display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.
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| <br>
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| [[Image:rga5.4.png|thumb|center|400px|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]
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| <br>
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| [[Image:rga5.5.png|thumb|center|400px|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{\lambda })= & 0.13519969 \\
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| & Var(\widehat{\beta })= & 0.017105343 \\
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| & Cov(\widehat{\beta },\widehat{\lambda })= & -0.046614609
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| \end{align}</math>
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|
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| '''Example 3'''
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| <br>
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| Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data in Table 5.1.
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| <br>
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| '''Solution'''
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| <br>
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| '''Fisher Matrix Bounds'''
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| <br>
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| From the previous example:
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| And for <math>T=620</math> hr, the partial derivatives of the cumulative and instantaneous MTBF are:
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| ::<math>\begin{align}
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| & \frac{\partial {{m}_{c}}(T)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }}{{T}^{1-\widehat{\beta }}}\ln T \\
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| & = & -\frac{1}{0.4239}{{620}^{0.3858}}\ln 620 \\
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| & = & -181.23135 \\
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| & \frac{\partial {{m}_{c}}(T)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}}{{T}^{1-\widehat{\beta }}} \\
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| & = & -\frac{1}{{{0.4239}^{2}}}{{620}^{0.3858}} \\
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| & = & -66.493299 \\
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| & \frac{\partial {{m}_{i}}(T)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }{{\widehat{\beta }}^{2}}}{{T}^{1-\beta }}-\frac{1}{\widehat{\lambda }\widehat{\beta }}{{T}^{1-\widehat{\beta }}}\ln T \\
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| & = & -\frac{1}{0.4239\cdot {{0.6142}^{2}}}{{620}^{0.3858}}-\frac{1}{0.4239\cdot 0.6142}{{620}^{0.3858}}\ln 620 \\
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| & = & -369.78634 \\
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| & \frac{\partial {{m}_{i}}(T)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}\widehat{\beta }}{{T}^{1-\widehat{\beta }}} \\
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| & = & -\frac{1}{{{0.4239}^{2}}\cdot 0.6142}\cdot {{620}^{0.3858}} \\
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| & = & -108.26001
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| \end{align}</math>
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| Therefore, the variances become:
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| ::<math>\begin{align}
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| & Var({{\widehat{m}}_{c}}(T))= & {{\left( -181.23135 \right)}^{2}}\cdot 0.017105343+{{\left( -66.493299 \right)}^{2}}\cdot 0.13519969 \\
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| & & -2\cdot \left( -181.23135 \right)\cdot \left( -66.493299 \right)\cdot 0.046614609 \\
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| & = & 36.113376
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| \end{align}</math>
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| ::<math>\begin{align}
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| & Var({{\widehat{m}}_{i}}(T))= & {{\left( -369.78634 \right)}^{2}}\cdot 0.017105343+{{\left( -108.26001 \right)}^{2}}\cdot 0.13519969 \\
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| & & -2\cdot \left( -369.78634 \right)\cdot \left( -108.26001 \right)\cdot 0.046614609 \\
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| & = & 191.33709
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| \end{align}</math>
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| So, at 90% confidence level and <math>T=620</math> hr, the Fisher Matrix confidence bounds are:
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| ::<math>\begin{align}
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| & {{[{{m}_{c}}(T)]}_{L}}= & {{{\hat{m}}}_{c}}(t){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
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| & = & 19.84581 \\
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| & {{[{{m}_{c}}(T)]}_{U}}= & {{{\hat{m}}}_{c}}(t){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
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| & = & 40.01927
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| \end{align}</math>
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| <br>
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| ::<math>\begin{align}
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| & {{[{{m}_{i}}(T)]}_{L}}= & {{{\hat{m}}}_{i}}(t){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
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| & = & 27.94261 \\
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| & {{[{{m}_{i}}(T)]}_{U}}= & {{{\hat{m}}}_{i}}(t){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
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| & = & 75.34193
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| \end{align}</math>
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| Figures 4fig86 and 4fig87 show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.
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| <br>
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| [[Image:rga5.6.png|thumb|center|400px|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]
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| <br>
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| <br>
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| [[Image:rga5.7.png|thumb|center|400px|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]
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|
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| <br>
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| '''Crow Bounds'''
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| <br>
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| The Crow confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% confidence level and for <math>T=620</math> hr are:
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| ::<math>\begin{align}
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| & {{[{{m}_{c}}(T)]}_{L}}= & \frac{1}{{{[{{\lambda }_{c}}(T)]}_{U}}} \\
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| & = & 20.5023 \\
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| & {{[{{m}_{c}}(T)]}_{U}}= & \frac{1}{{{[{{\lambda }_{c}}(T)]}_{L}}} \\
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| & = & 41.6282
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| \end{align}</math>
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| ::<math>\begin{align}
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| & {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\
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| & = & 30.7445 \\
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| & {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}} \\
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| & = & 84.7972
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| \end{align}</math>
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| Figures 4fig88 and 4fig89 show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.
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| <br>
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| [[Image:rga5.8.png|thumb|center|400px|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]
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| <br>
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| [[Image:rga5.9.png|thumb|center|400px|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]
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|
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| Confidence bounds can also be obtained on the parameters <math>\widehat{\beta }</math> and <math>\widehat{\lambda }</math> . For Fisher Matrix confidence bounds:
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| ::<math>\begin{align}
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| & {{\beta }_{L}}= & \hat{\beta }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
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| & = & 0.4325 \\
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| & {{\beta }_{U}}= & \hat{\beta }{{e}^{-{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
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| & = & 0.8722
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| \end{align}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\begin{align}
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| & {{\lambda }_{L}}= & \hat{\lambda }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
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| & = & 0.1016 \\
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| & {{\lambda }_{U}}= & \hat{\lambda }{{e}^{-{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
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| & = & 1.7691
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| \end{align}</math>
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| <br>
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| For Crow confidence bounds:
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| ::<math>\begin{align}
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| & {{\beta }_{L}}= & 0.4527 \\
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| & {{\beta }_{U}}= & 0.9350
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| \end{align}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\begin{align}
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| & {{\lambda }_{L}}= & 0.2870 \\
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| & {{\lambda }_{U}}= & 0.5827
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| \end{align}</math>
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