Template:Aae mle - Revision history

Richard House: Redirected page to Arrhenius Relationship#Parameter Estimation

2012-08-16T04:05:25Z

Redirected page to Arrhenius Relationship#Parameter Estimation

← Older revision		Revision as of 04:05, 16 August 2012
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	~~====Maximum Likelihood Estimation Method====~~		#REDIRECT [[Arrhenius_Relationship#Parameter_Estimation]]
	~~<br>~~
	~~The log-likelihood function for the exponential distribution is as shown next:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left~~[ ~~\lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \~~
	~~& \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]~~
	~~\end{align}</math>~~
	~~<br>~~
	~~where:~~

	~~<br>~~
	~~::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>~~


	~~<br>~~
	~~::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}</math>~~

	~~<br>~~
	~~and:~~
	~~<br>~~
	~~• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.~~

	~~• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.~~

	~~• <math>\lambda </math> is the failure rate parameter (unknown).~~

	~~• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.~~

	~~• <math>S</math> is the number of groups of suspension data points.~~

	~~• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.~~

	~~• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group.~~

	~~• <math>FI</math> is the number of interval data groups.~~

	~~• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.~~

	~~• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.~~

	~~• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.~~

	~~<br>~~
	~~Substituting the Arrhenius-exponential model into the log-likelihood function yields:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~\Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left~~[ ~~\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right~~] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	~~\end{align}</math>~~


	~~<br>~~
	~~where:~~

	~~<br>~~
	~~::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>~~

	~~<br>~~
	~~::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>~~


	~~<br>~~
	The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:


	~~<br>~~
	~~::<math>\begin{align}~~
	& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	~~\end{align}</math>~~


	~~<br>~~
	~~::<math>\begin{align}~~
	& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	~~\end{align}</math>~~

	~~<br>~~

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T17:34:19Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 17:34, 9 March 2012
Line 37:		Line 37:
	• <math>FI</math> is the number of interval data groups.		• <math>FI</math> is the number of interval data groups.

	• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.		• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.

	• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.		• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.

	• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.		• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.

	<br>		<br>

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-13T22:42:55Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 22:42, 13 February 2012
Line 48:		Line 48:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] \\		\Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	~~& &~~ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	\end{align}</math>		\end{align}</math>

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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\		& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	~~& &~~ \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	\end{align}</math>		\end{align}</math>

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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\		& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	~~& &~~ \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
	\end{align}</math>		\end{align}</math>

	<br>		<br>

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-13T22:40:57Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 22:40, 13 February 2012
Line 5:		Line 5:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \
	& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]		& \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
	\end{align}</math>		\end{align}</math>
	<br>		<br>
Line 42:		Line 42:

	• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.		• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.

	<br>		<br>
	Substituting the Arrhenius-exponential model into the log-likelihood function yields:		Substituting the Arrhenius-exponential model into the log-likelihood function yields:

Nicolette Young: Created page with '====Maximum Likelihood Estimation Method====
The log-likelihood function for the exponential distribution is as shown next:
:: $\begin{align} & \ln (L)= & \Lambd…'$

2012-01-12T19:09:52Z

Created page with '====Maximum Likelihood Estimation Method==== The log-likelihood function for the exponential distribution is as shown next: ::<math>\begin{align} & \ln (L)= & \Lambd…'

New page

====Maximum Likelihood Estimation Method====
 
The log-likelihood function for the exponential distribution is as shown next:

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}</math>
 
where:

 
::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}</math>

 
and:
 
• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.

• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.

• <math>\lambda </math> is the failure rate parameter (unknown).

• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.

• <math>S</math> is the number of groups of suspension data points.

• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.

• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group.

• <math>FI</math> is the number of interval data groups.

• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.

• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.

• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.
 
Substituting the Arrhenius-exponential model into the log-likelihood function yields:

 
::<math>\begin{align}
& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] \\
& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
\end{align}</math>

 
where:

 
::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>

 
The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}</math>

 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
& & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}</math>

Template:Aae mle - Revision history

Richard House: Redirected page to Arrhenius Relationship#Parameter Estimation

Chris Kahn: /* Maximum Likelihood Estimation Method */

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

Nicolette Young: Created page with '====Maximum Likelihood Estimation Method==== The log-likelihood function for the exponential distribution is as shown next: ::\begin{align} & \ln (L)= & \Lambd…'

Nicolette Young: Created page with '====Maximum Likelihood Estimation Method====
The log-likelihood function for the exponential distribution is as shown next:
:: $\begin{align} & \ln (L)= & \Lambd…'$