Normal Log-Likelihood Functions and their Partials - Revision history

Lisa Hacker: Redirected page to Appendix: Log-Likelihood Equations

2015-06-25T20:07:12Z

Redirected page to Appendix: Log-Likelihood Equations

← Older revision		Revision as of 20:07, 25 June 2015
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			#REDIRECT [[Appendix:_Log-Likelihood_Equations]]
	~~=== Normal Log-Likelihood Functions and their Partials===~~
	~~The complete normal likelihood function (without the constant) 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 }\phi \left( \frac{{{T}_{i}}-\mu }{\sigma } \right) \right] \\~~
	~~& +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{^{\prime }}\ln \left~~[ 1-~~\Phi \left( \frac{T_{i}^{^{\prime }}-\mu }{\sigma } \right) \right~~] \\
	& \text{ }+\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{^{\prime \prime }}\ln \left[ \Phi \left( \frac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right)-\Phi \left( \frac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \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 parameter (unknown a priori, the first of two parameters to be found)~~
	~~::• <math>\sigma </math> is the standard deviation parameter (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>{{F}_{i}}</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 }_{0}},{{\sigma }_{0}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0</math> and <math>\tfrac{\partial \Lambda }{\partial \sigma }=0.</math>

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

	~~::<math>\begin{align}~~
	\frac{\partial \Lambda }{\partial \sigma }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( {{T}_{i}}-\mu \right)}^{2}}}{{{\sigma }^{3}}}-\frac{1}{\sigma } \right) \\
	& +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\
	& -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}
	~~\end{align}</math>~~

	~~:where:~~

	~~::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{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>~~


	~~==== Complete Data====~~
	~~Note that for the normal distribution, and in the case of complete data only (as was shown in Chapter 3), there exists a closed-form solution for both of the parameters or:~~


	~~::<math>\widehat{\mu }=\widehat{{\bar{T}}}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{T}_{i}}</math>~~

	~~:and:~~

	~~::<math>\begin{align}~~
	~~\hat{\sigma }_{T}^{2}= & \frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}} \\~~
	~~{{{\hat{\sigma }}}_{T}}= & \sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}}}~~
	~~\end{align}</math>~~

Pantelis: Created page with ' === Normal Log-Likelihood Functions and their Partials=== The complete normal likelihood function (without the constant) is composed of three summation portions: :: $\begi…'$

2011-10-29T13:17:03Z

Created page with ' === Normal Log-Likelihood Functions and their Partials=== The complete normal likelihood function (without the constant) is composed of three summation portions: ::<math>\begi…'

New page

=== Normal Log-Likelihood Functions and their Partials===
The complete normal likelihood function (without the constant) 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 }\phi \left( \frac{{{T}_{i}}-\mu }{\sigma } \right) \right] \\
& +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{^{\prime }}\ln \left[ 1-\Phi \left( \frac{T_{i}^{^{\prime }}-\mu }{\sigma } \right) \right] \\
& \text{ }+\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{^{\prime \prime }}\ln \left[ \Phi \left( \frac{T_{{{R}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \right)-\Phi \left( \frac{T_{{{L}_{i}}}^{^{\prime \prime }}-\mu }{\sigma } \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 parameter (unknown a priori, the first of two parameters to be found)
::• <math>\sigma </math> is the standard deviation parameter (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>{{F}_{i}}</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 }_{0}},{{\sigma }_{0}} \right)</math> so that <math>\tfrac{\partial \Lambda }{\partial \mu }=0</math> and <math>\tfrac{\partial \Lambda }{\partial \sigma }=0.</math>

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

::<math>\begin{align}
\frac{\partial \Lambda }{\partial \sigma }= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( {{T}_{i}}-\mu \right)}^{2}}}{{{\sigma }^{3}}}-\frac{1}{\sigma } \right) \\
& +\frac{1}{\sigma }\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)}{1-\Phi \left( \tfrac{T_{i}^{\prime }-\mu }{\sigma } \right)} \\
& -\frac{1}{\sigma }\underset{i=1}{\overset{{{F}_{i}}}{\mathop \sum }}\,N_{i}^{\prime \prime }\frac{\left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)\phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}{\Phi \left( \tfrac{T_{Ri}^{\prime \prime }-\mu }{\sigma } \right)-\Phi \left( \tfrac{T_{Li}^{\prime \prime }-\mu }{\sigma } \right)}
\end{align}</math>

:where:

::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}{{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>

==== Complete Data====
Note that for the normal distribution, and in the case of complete data only (as was shown in Chapter 3), there exists a closed-form solution for both of the parameters or:

::<math>\widehat{\mu }=\widehat{{\bar{T}}}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{T}_{i}}</math>

:and:

::<math>\begin{align}
\hat{\sigma }_{T}^{2}= & \frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}} \\
{{{\hat{\sigma }}}_{T}}= & \sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\mathop \sum }}\,{{({{T}_{i}}-\bar{T})}^{2}}}
\end{align}</math>

Normal Log-Likelihood Functions and their Partials - Revision history

Lisa Hacker: Redirected page to Appendix: Log-Likelihood Equations

Pantelis: Created page with ' === Normal Log-Likelihood Functions and their Partials=== The complete normal likelihood function (without the constant) is composed of three summation portions: ::\begi…'

Pantelis: Created page with ' === Normal Log-Likelihood Functions and their Partials=== The complete normal likelihood function (without the constant) is composed of three summation portions: :: $\begi…'$