Appendix: Log-Likelihood Equations - Revision history

Lisa Hacker at 19:13, 15 September 2023

2023-09-15T19:13:34Z

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	{{Template:~~LDABOOK_SUB~~\|Appendix D\|Log-Likelihood Equations}}		{{Template:LDABOOK\|Appendix D\|Log-Likelihood Equations}}

	This appendix covers the log-likelihood functions and their associated partial derivatives for most of the distributions available in Weibull++. These distributions are discussed in more detail in the chapter for each distribution.		This appendix covers the log-likelihood functions and their associated partial derivatives for most of the distributions available in Weibull++. These distributions are discussed in more detail in the chapter for each distribution.

M Spivey: /* The Two-Parameter Exponential */

2022-11-14T20:27:18Z

The Two-Parameter Exponential

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	and		and

	::<math>\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda \,\!</math>.		::<math>\begin{align}\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda \,\!\end{align}</math>.

	From here we see that <math>\frac{\partial \Lambda }{\partial \gamma}</math> is a positive, constant function of <math>\gamma</math>. As alluded to in the ~~section~~ on the exponential distribution, this implies that the log-likelihood function <math>\Lambda</math> is, for fixed <math>\lambda</math>, an increasing function of <math>\gamma</math>. Thus the MLE for <math>\gamma</math> is its largest possible value <math>T_1</math>. Therefore, to find the full MLE solution <math>(\widehat{\lambda },\widehat{\gamma})</math> for the two-parameter exponential distribution, one should set <math>\gamma</math> equal to the first failure time and then find a <math>\lambda</math> such that <math>\tfrac{\partial \Lambda}{\partial \lambda} = 0</math>.		From here we see that <math>\frac{\partial \Lambda }{\partial \gamma}</math> is a positive, constant function of <math>\gamma</math>. As alluded to in the chapter on the exponential distribution, this implies that the log-likelihood function <math>\Lambda</math> is, for fixed <math>\lambda</math>, an increasing function of <math>\gamma</math>. Thus the MLE for <math>\gamma</math> is its largest possible value <math>T_1</math>. Therefore, to find the full MLE solution <math>(\widehat{\lambda },\widehat{\gamma})</math> for the two-parameter exponential distribution, one should set <math>\gamma</math> equal to the first failure time and then find (numerically) a <math>\lambda</math> such that <math>\tfrac{\partial \Lambda}{\partial \lambda} = 0</math>.

	The 3D Plot utility in Weibull++ further illustrates this behavior of the log-likelihood function, as shown next:		The 3D Plot utility in Weibull++ further illustrates this behavior of the log-likelihood function, as shown next:

M Spivey: /* The Two-Parameter Exponential */

2022-11-12T01:14:16Z

The Two-Parameter Exponential

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	:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval		:*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval

	~~The~~ two-parameter solution ~~will be found by solving for a pair of parameters (<math>\widehat{\lambda }~~,~~\widehat{\gamma }),\,\!</math> such that~~ <math>\tfrac{\partial \Lambda }{\partial \lambda }~~=0,\tfrac{\partial \Lambda }{\partial \gamma }=0.\,\!~~</math> ~~For the one-parameter case, solve for~~ <math>\tfrac{\partial \Lambda }{\partial \~~lambda~~ }~~=0.\,\!~~</math>		To find the two-parameter solution, look at the partial derivatives <math>\tfrac{\partial \Lambda }{\partial \lambda }</math> and <math>\tfrac{\partial \Lambda }{\partial \gamma}</math>:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	and:		and

	::<math>\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda \,\!</math>		::<math>\frac{\partial \Lambda }{\partial \gamma }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\lambda +\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime }\lambda \,\!</math>.

	~~Examination~~ the ~~derivative~~ for <math>\gamma\,\!</math> ~~will reveal~~ that:		From here we see that <math>\frac{\partial \Lambda }{\partial \gamma}</math> is a positive, constant function of <math>\gamma</math>. As alluded to in the section on the exponential distribution, this implies that the log-likelihood function <math>\Lambda</math> is, for fixed <math>\lambda</math>, an increasing function of <math>\gamma</math>. Thus the MLE for <math>\gamma</math> is its largest possible value <math>T_1</math>. Therefore, to find the full MLE solution <math>(\widehat{\lambda },\widehat{\gamma})</math> for the two-parameter exponential distribution, one should set <math>\gamma</math> equal to the first failure time and then find a <math>\lambda</math> such that <math>\tfrac{\partial \Lambda}{\partial \lambda} = 0</math>.

	::<math>\frac{\partial \Lambda }{\partial \gamma }=\left( \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime } \right)\lambda \equiv 0\,\!</math>		The 3D Plot utility in Weibull++ further illustrates this behavior of the log-likelihood function, as shown next:

	~~The above equation will be equal to zero only if either:~~

	~~::<math>\lambda =0\,\!</math>~~

	~~or:~~

	::<math>\left( \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ \ +\underset{i=1}{\overset{FI}{\mathop \sum }}\,N_{i}^{\prime \prime } \right)=0\,\!</math>

	This is an unwelcome fact, alluded to earlier in the chapter, that essentially indicates that there is no realistic solution for the two-parameter MLE for exponential. The above equations indicate that there is no non-trivial MLE solution that satisfies both <math>\tfrac{\partial \Lambda }{\partial \lambda }=0,\tfrac{\partial \Lambda }{\partial \gamma }=0.\,\!</math>
	It can be shown that the best solution for <math>\gamma ,\,\!</math> satisfying the constraint that <math>\gamma \le {{T}_{1}}\,\!</math> is <math>\gamma ={{T}_{1}}.\,\!</math> To then solve for the two-parameter exponential distribution via MLE, one can set equal to the first time-to-failure, and then find a <math>\lambda \,\!</math> such that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0.\,\!</math>

	Using this methodology, a maximum can be achieved along the <math>\lambda \,\!</math>-axis, and a local maximum along the <math>\gamma \,\!</math>-axis at <math>\gamma ={{T}_{1}}\,\!</math>, constrained by the fact that <math>\gamma \le {{T}_{1}}\,\!</math>. The 3D Plot utility in Weibull++ illustrates this behavior of the log-likelihood function, as shown next:

	[[image: appendixc__127.gif\|center\|350px]]		[[image: appendixc__127.gif\|center\|350px]]

Sharon Honecker: /* Lognormal Log-Likelihood Functions and their Partials */

2016-01-18T22:54:07Z

Lognormal Log-Likelihood Functions and their Partials

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	where:		where:

	::<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{{{x}^{2}}}{2}}}\,\!</math>

	and:		and:

Sharon Honecker: /* Normal Log-Likelihood Functions and their Partials */

2016-01-18T22:53:16Z

Normal Log-Likelihood Functions and their Partials

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	where:		where:

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

	and:		and:

Harry Guo: /* Lognormal Log-Likelihood Functions and their Partials */

2013-05-17T22:35:14Z

Lognormal Log-Likelihood Functions and their Partials

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

Harry Guo: /* Lognormal Log-Likelihood Functions and their Partials */

2013-05-17T22:34:10Z

Lognormal Log-Likelihood Functions and their Partials

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	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>:		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}		::<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)} \\		& +\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)}		& \ \ -\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>		\end{align}\,\!</math>

Harry Guo: /* Lognormal Log-Likelihood Functions and their Partials */

2013-05-17T22:32:58Z

Lognormal Log-Likelihood Functions and their Partials

← Older revision		Revision as of 22:32, 17 May 2013
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	::<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)} \\		& +\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)}		& \ \ -\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)}

Harry Guo: /* Lognormal Log-Likelihood Functions and their Partials */

2013-05-17T22:31:59Z

Lognormal Log-Likelihood Functions and their Partials

← Older revision		Revision as of 22:31, 17 May 2013
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	::<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)} \\		& +\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)}		& \ \ -\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)}

Harry Guo: /* Lognormal Log-Likelihood Functions and their Partials */

2013-05-17T22:31:21Z

Lognormal Log-Likelihood Functions and their Partials

← Older revision		Revision as of 22:31, 17 May 2013
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	::<math>\begin{align}		::<math>\begin{align}
	\frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}~~\frac{{~~{N}_{i~~}}}{{{T}_{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)} \\		& +\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)}		& \ \ -\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)}