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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:
 
::<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>
 
And the probability of success (reliability) for each configuration  <math>i</math>  is equal to:
 
::<math>{{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}</math>
 
The likelihood function is:
<br>
::<math>\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}
  {{N}_{i}}  \\
  {{M}_{i}}  \\
\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>
 
Taking the natural log on both sides yields:
 
::<math>\begin{align}
  & \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}
  {{N}_{i}}  \\
  {{M}_{i}}  \\
\end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\
&  & +\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] 
\end{align}</math>
 
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:
 
::<math>\begin{align}
  & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\
& \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0 
\end{align}</math>
 
:where:
 
::<math>\begin{align}
  & {{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] \\
& {{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]} \\
& {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\, 
\end{align}</math>
 
'''Example 8'''
<br>
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.
<br>
• Configuration 1 experienced 5 failures,
<br>
• Configuration 2 experienced 3 failures,
<br>
• Configuration 3 experienced 4 failures and
<br>
• Configuration 4 experienced 4 failures.
<br>
<br>
:1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
:2) Estimate the unreliability and reliability by configuration.
<br>
<br>
'''Solution'''
<br>
: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.
:2) Table 5.6 displays the results of Eqns. (ffffi) and (rrrri).
Figures 4fig816 and 4fig817 show plots of the estimated unreliability and reliability by configuration.
<br>
<center>Table 5.6 - Estimated failure probability and reliability by configuration</center>
{|style= align="center" border="2"
!Configuration(<math>i</math>)
!Estimated Failure Probability
!Estimated Reliability
|-
|1|| 0.333|| 0.667
|-
|2|| 0.234|| 0.766
|-
|3|| 0.206|| 0.794
|-
|4|| 0.190|| 0.810
|}
<br>
[[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
<br>
[[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]]
<br>

Latest revision as of 23:34, 23 August 2012

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