Template:Ald mle - Revision history

Chris Kahn: /* MLE Parameter Estimation */

2012-03-07T17:02:24Z

MLE Parameter Estimation

← Older revision		Revision as of 17:02, 7 March 2012
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	<br>		<br>
	The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> where:		The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0</math>, where:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}

Chris Kahn: /* MLE Parameter Estimation */

2012-03-07T17:01:30Z

MLE Parameter Estimation

Athanasios Gerokostopoulos: /* MLE Parameter Estimation */

2012-02-27T23:32:37Z

MLE Parameter Estimation

← Older revision		Revision as of 23:32, 27 February 2012
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	:• <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 i <math>^{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 i <math>^{th}</math> interval.

			<br>
	The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> where:		The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> where:
	<br>		<br>
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	\end{align}</math>		\end{align}</math>
	<br>		<br>
	:and:		and:
	<br>		<br>
	<br>		<br>

Athanasios Gerokostopoulos: /* MLE Parameter Estimation */

2012-02-27T23:32:15Z

MLE Parameter Estimation

← Older revision		Revision as of 23:32, 27 February 2012
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	<br>		<br>
	:where:		where:
	<br>		<br>
	::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>		::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>
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	::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>		::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>
	<br>		<br>
	:and:		and:
	<br>		<br>
	<br>		<br>

Athanasios Gerokostopoulos: /* MLE Parameter Estimation */

2012-02-07T01:09:54Z

MLE Parameter Estimation

← Older revision		Revision as of 01:09, 7 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\		& \ln (L)= \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]\text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]+\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 })]
	~~& &~~ \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]+\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>

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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-{\mu }') \\		& \frac{\partial \Lambda }{\partial {\mu }'}= \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-{\mu }')+\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
	~~& &~~ +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
	& & \\		& & \\
	& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\		& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right)+\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
	~~& &~~ +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
	\end{align}</math>		\end{align}</math>
	<br>		<br>
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	::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>		::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>
	<br>		<br>
	::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dx</math>		::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dx</math>
	<br>		<br>

Nicolette Young: Created page with '====MLE Parameter Estimation==== The parameters of the lognormal distribution can also be estimated using Maximum Likelihood Estimation (MLE). This general log-likelihood functio…'

2012-01-12T18:54:36Z

Created page with '====MLE Parameter Estimation==== The parameters of the lognormal distribution can also be estimated using Maximum Likelihood Estimation (MLE). This general log-likelihood functio…'

New page

====MLE Parameter Estimation====
The parameters of the lognormal distribution can also be estimated using Maximum Likelihood Estimation (MLE). This general log-likelihood function is:

 
::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right] \\
& & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]+\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>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>
 
::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>
 
:and:
 
 
:• <math>{{F}_{e}}</math> is the number of groups of 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>{\mu }'</math> is the mean of the natural logarithms of the times-to-failure (unknown a priori, the first of two parameters to be found).
:• <math>{{\sigma }_{{{T}'}}}</math> is the standard deviation of the natural logarithms of the times-to-failure (unknown a priori, the second of two parameters to be found).
:• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data.
:• <math>S</math> is the number of groups of suspension data points.
:• <math>N_{i}^{\prime }</math> is the number of suspensions in <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.
The solution will be found by solving for a pair of parameters <math>\left( {\mu }',{{\sigma }_{{{T}'}}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial {\mu }'}=0</math> and <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math> where:
 
::<math>\begin{align}
& \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-{\mu }') \\
& & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
& & \\
& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-{\mu }' \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
& & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}</math>
 
:and:
 
 
::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>
 
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dx</math>

← Older revision		Revision as of 17:01, 7 March 2012
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	====MLE Parameter Estimation====		====MLE Parameter Estimation====
	The parameters of the lognormal distribution can also be estimated using ~~Maximum Likelihood Estimation~~ (MLE). This general log-likelihood function is:		The parameters of the lognormal distribution can also be estimated using maximum likelihood estimation (MLE). This general log-likelihood function is:

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

	<br>		<br>
	::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>		::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-{\mu }'}{\sigma _{T}^{\prime }}</math>
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	<br>		<br>
	<br>		<br>
	:• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points.		:* <math>{{F}_{e}}</math> is the number of groups of 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>{{N}_{i}}</math> is the number of failure times in the <math>{{i}^{th}}</math> time-to-failure data group.
	:• <math>{\mu }'</math> is the mean of the natural logarithms of the times~~-to-failure~~ (unknown a priori, the first of two parameters to be found).		:* <math>{\mu }'</math> is the mean of the natural logarithms of the failure times (unknown a priori, the first of two parameters to be found).
	:• <math>{{\sigma }_{{{T}'}}}</math> is the standard deviation of the natural logarithms of the times~~-to-failure~~ (unknown a priori, the second of two parameters to be found).		:* <math>{{\sigma }_{{{T}'}}}</math> is the standard deviation of the natural logarithms of the failure times (unknown a priori, the second of two parameters to be found).
	:• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data.		:* <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data.
	:• <math>S</math> is the number of groups of suspension data points.		:* <math>S</math> is the number of groups of suspension data points.
	:• <math>N_{i}^{\prime }</math> is the number of suspensions in <math>{{i}^{th}}</math> group of suspension data points.		:* <math>N_{i}^{\prime }</math> is the number of suspensions in <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>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>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>