Template:Loglogistic distribution - Revision history

Lisa Hacker: Redirected page to The Loglogistic Distribution

2012-08-09T09:58:11Z

Redirected page to The Loglogistic Distribution

← Older revision		Revision as of 09:58, 9 August 2012
Line 1:		Line 1:
	~~{{loglogistic distribution introduction}}~~		#REDIRECT [[The_Loglogistic_Distribution]]

	~~{{loglogistic probability density function}}~~

	~~{{loglogistic mean median and mode}}~~

	~~{{loglogistic standard deviation}}~~

	~~{{loglogistic reliability function}}~~

	~~{{loglogistic reliable life}}~~

	~~{{loglogistic failure rate function}}~~

	~~{{loglogistic distribution characteristics}}~~

	~~{{loglogistic confidence bounds}}~~

	~~==General Examples==~~
	~~'''Example 1:'''~~
	~~{{Example: LogLogistic distribution example}}~~

Harry Guo at 18:03, 20 February 2012

2012-02-20T18:03:23Z

← Older revision		Revision as of 18:03, 20 February 2012
Line 19:		Line 19:
	==General Examples==		==General Examples==
	'''Example 1:'''		'''Example 1:'''
	{{~~loglogistic~~ example}}		{{Example: LogLogistic distribution example}}

Harry Guo at 18:01, 20 February 2012

2012-02-20T18:01:50Z

← Older revision		Revision as of 18:01, 20 February 2012
Line 17:		Line 17:
	{{loglogistic confidence bounds}}		{{loglogistic confidence bounds}}

	==General Examples===		==General Examples==
	'''Example 1:'''		'''Example 1:'''
	{{loglogistic example}}		{{loglogistic example}}

Harry Guo at 18:01, 20 February 2012

2012-02-20T18:01:41Z

← Older revision		Revision as of 18:01, 20 February 2012
Line 17:		Line 17:
	{{loglogistic confidence bounds}}		{{loglogistic confidence bounds}}

			==General Examples===
	'''Example 1:'''		'''Example 1:'''
	{{loglogistic example}}		{{loglogistic example}}

Harry Guo at 18:00, 20 February 2012

2012-02-20T18:00:17Z

← Older revision		Revision as of 18:00, 20 February 2012
Line 17:		Line 17:
	{{loglogistic confidence bounds}}		{{loglogistic confidence bounds}}

			'''Example 1:'''
	{{loglogistic example}}		{{loglogistic example}}

Harry Guo: /* The Loglogistic Distribution */

2012-02-15T01:16:09Z

The Loglogistic Distribution

← Older revision		Revision as of 01:16, 15 February 2012
Line 1:		Line 1:
	~~===The Loglogistic Distribution===~~		{{loglogistic distribution introduction}}
	~~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 [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}}

Nicolette Young: /* A LogLogistic Distribution Example */

2012-01-19T22:33:28Z

A LogLogistic Distribution Example

← Older revision		Revision as of 22:33, 19 January 2012
Line 18:		Line 18:
	{{loglogistic confidence bounds}}		{{loglogistic confidence bounds}}

	~~====A LogLogistic Distribution Example====~~		{{loglogistic example}}
	~~Determine the loglogistic parameter estimates for the data given in Table 10.3.~~

	~~<center><math>\overset~~{{}}~~{\mathop{\text{Table 10}\text{.3 - Test data}}}\,</math></center>~~

	~~<center><math>\begin{matrix}~~
	~~\text{Data point index} & \text{Last Inspected} & \text{State End time} \\~~
	~~\text{1} & \text{105} & \text{106} \\~~
	~~\text{2} & \text{197} & \text{200} \\~~
	~~\text{3} & \text{297} & \text{301} \\~~
	~~\text{4} & \text{330} & \text{335} \\~~
	~~\text{5} & \text{393} & \text{401} \\~~
	~~\text{6} & \text{423} & \text{426} \\~~
	~~\text{7} & \text{460} & \text{468} \\~~
	~~\text{8} & \text{569} & \text{570} \\~~
	~~\text{9} & \text{675} & \text{680} \\~~
	~~\text{10} & \text{884} & \text{889} \\~~
	~~\end{matrix}</math></center>~~


	Using Times-to-failure data under the Folio Data Type and the My data set contains interval and/or left censored data under Times-to-failure data options to enter the above data, the computed parameters for maximum likelihood are calculated to be:

	~~::<math>\begin{align}~~
	~~& {{{\hat{\mu }}}^{\prime }}= & 5.9772 \\~~
	~~& {{{\hat{\sigma }}}_{{{T}'}}}= & 0.3256~~
	~~\end{align}</math>~~


	~~For rank regression on <math>X\ \ :</math>~~

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 5.9281 \\~~
	~~& \hat{\sigma }= & 0.3821~~
	~~\end{align}</math>~~


	~~For rank regression on <math>Y\ \ :</math>~~

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 5.9772 \\~~
	~~& \hat{\sigma }= & 0.3256~~
	~~\end{align}</math>~~

Nicolette Young: /* Bounds on Time */

2012-01-04T22:33:21Z

Bounds on Time

← Older revision		Revision as of 22:33, 4 January 2012
Line 17:		Line 17:

	{{loglogistic confidence bounds}}		{{loglogistic confidence bounds}}

	~~====Bounds on Time====~~
	~~The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:~~

	~~::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}}</math>~~


	~~:where:~~

	~~::<math>z=\ln (1-R)-\ln (R)</math>~~


	~~:or:~~

	~~::<math>\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>~~


	~~:Let:~~

	~~::<math>u=\ln (T)=\widehat{\mu }+\widehat{\sigma }z</math>~~


	~~:then:~~

	~~::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>~~




	~~::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ }</math>~~


	~~:where:~~


	::<math>Var(\widehat{u})={{(\frac{\partial u}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial u}{\partial \mu })(\frac{\partial u}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial u}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>


	~~:or:~~

	~~::<math>Var(\widehat{u})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>~~


	~~The upper and lower bounds are then found by:~~

	~~::<math>{{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)}</math>~~



	~~::<math>{{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)}</math>~~

	====A LogLogistic Distribution Example====		====A LogLogistic Distribution Example====

Nicolette Young: /* Confidence Bounds */

2012-01-04T22:32:55Z

Confidence Bounds

@@ Line 16: / Line 16: @@
 {{loglogistic distribution characteristics}}
-====Confidence Bounds====
+{{loglogistic confidence bounds}}
 ====Bounds on Time====

Nicolette Young: /* Distribution Characteristics */

2012-01-04T22:32:23Z

Distribution Characteristics

@@ Line 14: / Line 14: @@
 {{loglogistic failure rate function}}
-====Distribution Characteristics====
+{{loglogistic distribution characteristics}}
 ====Confidence Bounds====