The Loglogistic Distribution - Revision history

Nicolette Young: /* Distribution Characteristics */

2015-12-23T17:12:04Z

Distribution Characteristics

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	:* <math>\lambda (t)\,\!</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}\,\!</math> and decreases thereafter. <math>\lambda (t)\,\!</math> is concave at first, then becomes convex.		:* <math>\lambda (t)\,\!</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}\,\!</math> and decreases thereafter. <math>\lambda (t)\,\!</math> is concave at first, then becomes convex.

	[[Image:WB.15 loglogistic pdf.png\|center\|~~350px~~\| ]]		[[Image:WB.15 loglogistic pdf.png\|center\|400px\| ]]

	==Confidence Bounds==		==Confidence Bounds==

Lisa Hacker at 23:51, 23 June 2015

2015-06-23T23:51:55Z

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	==General Examples==		==General Examples==
	{{:Loglogistic Distribution Example}}		{{:Loglogistic Distribution Example}}


	~~[[Category: Completed Theoretical Review]]~~

Richard House at 17:55, 26 September 2012

2012-09-26T17:55:18Z

Show changes

Richard House at 17:40, 5 September 2012

2012-09-05T17:40:30Z

← Older revision		Revision as of 17:40, 5 September 2012
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	{{template:LDABOOK\|15\|The LogLogistic Distribution}}		{{template:LDABOOK\|15\|The LogLogistic Distribution}}
	As may be indicated by the name, the loglogistic distribution has certain similarities to the logistic distribution. A random variable is loglogistically distributed if the logarithm of the random variable is logistically distributed. Because of this, there are many mathematical similarities between the two distributions [[Appendix: ~~Weibull References~~\|[27]]]. For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.		As may be indicated by the name, the loglogistic distribution has certain similarities to the logistic distribution. A random variable is loglogistically distributed if the logarithm of the random variable is logistically distributed. Because of this, there are many mathematical similarities between the two distributions, as discussed in Meeker and Escobar [[Appendix:_Life_Data_Analysis_References\|[27]]]. For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.

	===Loglogistic Probability Density Function===		===Loglogistic Probability Density Function===

Richard House at 17:39, 5 September 2012

2012-09-05T17:39:36Z

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	===Loglogistic Probability Density Function===		===Loglogistic Probability Density Function===
	The loglogistic distribution is a 2-parameter distribution with parameters <math>\mu \,\!</math> and <math>\sigma \,\!</math> . The ~~<math>~~pdf~~</math>~~ for this distribution is given by:		The loglogistic distribution is a 2-parameter distribution with parameters <math>\mu \,\!</math> and <math>\sigma \,\!</math> . The ''pdf'' for this distribution is given by:

	::<math>f(t)=\frac{{{e}^{z}}}{\sigma {t}{{(1+{{e}^{z}})}^{2}}}</math>		::<math>f(t)=\frac{{{e}^{z}}}{\sigma {t}{{(1+{{e}^{z}})}^{2}}}</math>
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	:* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.		:* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
	:* The ~~<math>~~pdf~~</math>~~ starts at zero, increases to its mode, and decreases thereafter.		:* The ''pdf'' starts at zero, increases to its mode, and decreases thereafter.
	:* As <math>\mu </math> increases, while <math>\sigma \,\!</math> is kept the same, the ~~<math>~~pdf~~</math>~~ gets stretched out to the right and its height decreases, while maintaining its shape.		:* As <math>\mu </math> increases, while <math>\sigma \,\!</math> is kept the same, the ''pdf'' gets stretched out to the right and its height decreases, while maintaining its shape.
	:* As <math>\mu </math> decreases,while <math>\sigma \,\!</math> is kept the same, the ~~<math>~~pdf~~</math>~~ gets pushed in towards the left and its height increases.		:* As <math>\mu </math> decreases,while <math>\sigma \,\!</math> is kept the same, the ''pdf'' gets pushed in towards the left and its height increases.
	:* <math>\lambda (t)\,\!</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}\,\!</math> and decreases thereafter. <math>\lambda (t)\,\!</math> is concave at first, then becomes convex.		:* <math>\lambda (t)\,\!</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}\,\!</math> and decreases thereafter. <math>\lambda (t)\,\!</math> is concave at first, then becomes convex.

Richard House: /* The Standard Deviation */

2012-08-28T21:27:10Z

The Standard Deviation

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	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation of the loglogistic distribution, <math>{{\sigma }_{T}}</math> , is given by:		The standard deviation of the loglogistic distribution, <math>{{\sigma }_{T}}\,\!</math> , is given by:

	::<math>{{\sigma }_{T}}={{e}^{\mu }}\sqrt{\Gamma (1+2\sigma )\Gamma (1-2\sigma )-{{(\Gamma (1+\sigma )\Gamma (1-\sigma ))}^{2}}}</math>		::<math>{{\sigma }_{T}}={{e}^{\mu }}\sqrt{\Gamma (1+2\sigma )\Gamma (1-2\sigma )-{{(\Gamma (1+\sigma )\Gamma (1-\sigma ))}^{2}}}</math>

Richard House: /* Bounds on Reliability */

2012-08-28T21:25:44Z

Bounds on Reliability

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	::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math>		::<math>\widehat{z}=\frac{{t}'-\widehat{\mu }}{\widehat{\sigma }}</math>

	Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<t'=\ln (t)<\infty </math> and <math>z</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .		Here <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> , <math>0<\sigma <\infty </math> , therefore <math>0<t'=\ln (t)<\infty </math> and <math>z\,\!</math> also is changing from <math>-\infty </math> till <math>+\infty </math> .

	The bounds on <math>z</math> are estimated from:		The bounds on <math>z\,\!</math> are estimated from:

	::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>		::<math>{{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})}</math>
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	::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)}</math>		::<math>{{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)}</math>


	===Bounds on Time===		===Bounds on Time===

Richard House: /* Bounds on the Parameters */

2012-08-28T21:25:11Z

Bounds on the Parameters

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	\end{align}</math>		\end{align}</math>

	where <math>{{K}_{\alpha }}</math> is defined by:		where <math>{{K}_{\alpha }}\,\!</math> is defined by:

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

	If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds.		If <math>\delta \,\!</math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}\,\!</math> for the two-sided bounds, and <math>\alpha =1-\delta \,\!</math> for the one-sided bounds.

	The variances and covariances of <math>\widehat{\mu }</math> and <math>\widehat{\sigma }</math> are estimated as follows:		The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{\sigma }\,\!</math> are estimated as follows:

	::<math>\left( \begin{matrix}		::<math>\left( \begin{matrix}
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	\end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1}</math>		\end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1}</math>

	where <math>\Lambda </math> is the log-likelihood function of the loglogistic distribution.		where <math>\Lambda \,\!</math> is the log-likelihood function of the loglogistic distribution.

	===Bounds on Reliability===		===Bounds on Reliability===

Richard House: /* Distribution Characteristics */

2012-08-28T21:24:22Z

Distribution Characteristics

Richard House: /* The Loglogistic Reliability Function */

2012-08-28T21:21:07Z

The Loglogistic Reliability Function

← Older revision		Revision as of 21:21, 28 August 2012
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	===The Loglogistic Reliability Function===		===The Loglogistic Reliability Function===
	The reliability for a mission of time <math>T</math> , starting at age 0, for the loglogistic distribution is determined by:		The reliability for a mission of time <math>T\,\!</math> , starting at age 0, for the loglogistic distribution is determined by:

	::<math>R=\frac{1}{1+{{e}^{z}}}</math>		::<math>R=\frac{1}{1+{{e}^{z}}}</math>

← Older revision		Revision as of 21:24, 28 August 2012
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	==Distribution Characteristics==		==Distribution Characteristics==
	For <math>\sigma >1</math> :		For <math>\sigma >1\,\!</math> :

	:* <math>f(t)</math> decreases monotonically and is convex. Mode and mean do not exist.		:* <math>f(t)\,\!</math> decreases monotonically and is convex. Mode and mean do not exist.

	For <math>\sigma =1</math> :		For <math>\sigma =1\,\!</math> :

	:* <math>f(t)</math> decreases monotonically and is convex. Mode and mean do not exist. As <math>t\to 0</math> , <math>f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>		:* <math>f(t)\,\!</math> decreases monotonically and is convex. Mode and mean do not exist. As <math>t\to 0\,\!</math> , <math>f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.\,\!</math>
	:* As <math>t\to 0</math> , <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>		:* As <math>t\to 0\,\!</math> , <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.\,\!</math>

	For <math>0<\sigma <1</math> :		For <math>0<\sigma <1\,\!</math> :

	:* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.		:* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
	:* The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.		:* The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.
	:* As <math>\mu </math> increases, while <math>\sigma </math> is kept the same, the <math>pdf</math> gets stretched out to the right and its height decreases, while maintaining its shape.		:* As <math>\mu </math> increases, while <math>\sigma \,\!</math> is kept the same, the <math>pdf</math> gets stretched out to the right and its height decreases, while maintaining its shape.
	:* As <math>\mu </math> decreases,while <math>\sigma </math> is kept the same, the .. gets pushed in towards the left and its height increases.		:* As <math>\mu </math> decreases,while <math>\sigma \,\!</math> is kept the same, the <math>pdf</math> gets pushed in towards the left and its height increases.
	:* <math>\lambda (t)</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}</math> and decreases thereafter. <math>\lambda (t)</math> is concave at first, then becomes convex.		:* <math>\lambda (t)\,\!</math> increases till <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}\,\!</math> and decreases thereafter. <math>\lambda (t)\,\!</math> is concave at first, then becomes convex.

	[[Image:WB.15 loglogistic pdf.png\|center\|350px\| ]]		[[Image:WB.15 loglogistic pdf.png\|center\|350px\| ]]