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| ===Maximum Likelihood Estimators===
| | #REDIRECT [[Crow-AMSAA - NHPP]] |
| This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illustrating these concepts. The estimation procedures described below provide maximum likelihood estimates (MLEs) for the model's two parameters, <math>\lambda </math> and <math>\beta </math> . The MLEs for <math>\lambda </math> and <math>\beta </math> allow for point estimates for the probability of failure, given by:
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| ::<math>{{\hat{f}}_{i}}=\frac{\hat{\lambda }T_{i}^{{\hat{\beta }}}-\hat{\lambda }T_{i-1}^{{\hat{\beta }}}}{{{N}_{i}}}=\frac{\hat{\lambda }\left( T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}} \right)}{{{N}_{i}}}</math>
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| And the probability of success (reliability) for each configuration <math>i</math> is equal to:
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| ::<math>{{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}</math>
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| The likelihood function is:
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| <br>
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| ::<math>\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}
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| {{N}_{i}} \\
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| {{M}_{i}} \\
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| \end{matrix} \right){{\left( \frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}} \right)}^{{{M}_{i}}}}{{\left( \frac{{{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta }}{{{N}_{i}}} \right)}^{{{N}_{i}}-{{M}_{i}}}}</math>
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| Taking the natural log on both sides yields:
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| ::<math>\begin{align}
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| & \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}
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| {{N}_{i}} \\
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| {{M}_{i}} \\
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| \end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\
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| & & +\underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ ({{N}_{i}}-{{M}_{i}})\left[ \ln ({{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right]
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| \end{align}</math>
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| Taking the derivative with respect to <math>\lambda </math> and <math>\beta </math> respectively, exact MLEs for <math>\lambda </math> and <math>\beta </math> are values satisfying the following two equations:
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| ::<math>\begin{align}
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| & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\
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| & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0
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| \end{align}</math>
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| :where:
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| ::<math>\begin{align}
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| & {{H}_{i}}= & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ T_{i}^{\beta }\ln {{T}_{i}}-T_{i-1}^{\beta }\ln {{T}_{i-1}} \right] \\
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| & {{S}_{i}}= & \frac{{{M}_{i}}}{\left[ \lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta } \right]}-\frac{{{N}_{i}}-{{M}_{i}}}{\left[ {{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta } \right]} \\
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| & {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\,
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| \end{align}</math>
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| '''Example 8'''
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| <br>
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| A one-shot system underwent reliability growth development testing for a total of 68 trials. Delayed corrective actions were incorporated after the 14th, 33rd and 48th trials. From trial 49 to trial 68, the configuration was not changed.
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| <br>
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| • Configuration 1 experienced 5 failures,
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| <br>
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| • Configuration 2 experienced 3 failures,
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| <br>
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| • Configuration 3 experienced 4 failures and
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| <br>
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| • Configuration 4 experienced 4 failures.
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| <br>
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| <br>
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| :1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
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| :2) Estimate the unreliability and reliability by configuration.
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| <br>
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| <br>
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| '''Solution'''
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| <br>
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| :1) The solution of Eqns. (solution1) and (solution2) provides for <math>\lambda </math> and <math>\beta </math> corresponding to 0.5954 and 0.7801, respectively.
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| :2) Table 5.6 displays the results of Eqns. (ffffi) and (rrrri).
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| Figures 4fig816 and 4fig817 show plots of the estimated unreliability and reliability by configuration.
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| <br>
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| <center>Table 5.6 - Estimated failure probability and reliability by configuration</center>
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| {|style= align="center" border="2"
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| !Configuration(<math>i</math>)
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| !Estimated Failure Probability
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| !Estimated Reliability
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| |-
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| |1|| 0.333|| 0.667
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| |-
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| |2|| 0.234|| 0.766
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| |-
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| |3|| 0.206|| 0.794
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| |-
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| |4|| 0.190|| 0.810
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| |}
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| <br>
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| [[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
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| <br>
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| [[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]]
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| <br>
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