Lisa Hacker: Redirected page to Crow-AMSAA - NHPP

2012-08-23T23:34:22Z

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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>~~

Nicolette Young: Created page with '===Maximum Likelihood Estimators=== This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illu…'

2012-01-09T17:12:05Z

Created page with '===Maximum Likelihood Estimators=== This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illu…'

New page

===Maximum Likelihood Estimators===
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:
 
::<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'''
 
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.
 
• Configuration 1 experienced 5 failures,
 
• Configuration 2 experienced 3 failures,
 
• Configuration 3 experienced 4 failures and
 
• Configuration 4 experienced 4 failures.
 
 
:1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
:2) Estimate the unreliability and reliability by configuration.
 
 
'''Solution'''
 
: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.
 
<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
|}
 
[[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
 
[[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]]

Template:Maximum likelihood estimators camsaa-cd - Revision history

Lisa Hacker: Redirected page to Crow-AMSAA - NHPP

Nicolette Young: Created page with '===Maximum Likelihood Estimators=== This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illu…'