Lognormal Log-Likelihood Functions and their Partials - Revision history

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2015-06-25T20:07:49Z

Redirected page to Appendix: Log-Likelihood Equations

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	#~~REDIRCT~~ [[Appendix:_Log-Likelihood_Equations]]		#REDIRECT [[Appendix:_Log-Likelihood_Equations]]

Lisa Hacker: Replaced content with '#REDIRCT Appendix:_Log-Likelihood_Equations'

2015-06-25T20:07:40Z

Replaced content with '#REDIRCT Appendix:_Log-Likelihood_Equations'

← Older revision		Revision as of 20:07, 25 June 2015
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	~~=== Lognormal Log-Likelihood Functions and their Partials===~~		#REDIRCT [[Appendix:_Log-Likelihood_Equations]]
	~~The general log-likelihood function (without the constant) for the lognormal distribution is composed of three summation portions:~~


	~~::<math>\begin{align}~~
	~~\ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left~~[ ~~\frac{1}{{{\sigma }_{{{T}'}}}}\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~~] \\
	& \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \Phi \left( \frac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \frac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]
	~~\end{align}</math>~~

	~~:where:~~
	~~::• <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 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~~
	~~::• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{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>:

	~~::<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)} \\
	& \ \ -\underset{i=1}{\overset{FI}{\mathop \sum }}\,\frac{N_{i}^{\prime \prime }}{\sigma }\frac{\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}
	~~\end{align}</math>~~
	~~<math></math>~~

	~~:where:~~

	~~::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>~~

	~~:and:~~

	~~::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>~~
	~~<br>~~

Pantelis: Created page with '=== Lognormal Log-Likelihood Functions and their Partials=== The general log-likelihood function (without the constant) for the lognormal distribution is composed of three summa…'

2011-10-29T13:17:40Z

Created page with '=== Lognormal Log-Likelihood Functions and their Partials=== The general log-likelihood function (without the constant) for the lognormal distribution is composed of three summa…'

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=== Lognormal Log-Likelihood Functions and their Partials===
The general log-likelihood function (without the constant) for the lognormal distribution is composed of three summation portions:

::<math>\begin{align}
\ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}}\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] \\
& \text{ }+\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\ln \left[ \Phi \left( \frac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \frac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right) \right]
\end{align}</math>

:where:
::• <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 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
::• and <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{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>:

::<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)} \\
& \ \ -\underset{i=1}{\overset{FI}{\mathop \sum }}\,\frac{N_{i}^{\prime \prime }}{\sigma }\frac{\phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}{\Phi \left( \tfrac{\ln \left( T_{Ri}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)-\Phi \left( \tfrac{\ln \left( T_{Li}^{\prime \prime } \right)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)}
\end{align}</math>
<math></math>

:where:

::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>

:and:

::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
<br>