Template:GeneralizedGammaDistribution - Revision history

Lisa Hacker: Redirected page to The Generalized Gamma Distribution

2012-08-09T09:33:47Z

Redirected page to The Generalized Gamma Distribution

← Older revision		Revision as of 09:33, 9 August 2012
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	[[~~Category: For Deletion~~]]		#REDIRECT [[The_Generalized_Gamma_Distribution]]

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-07-20T12:20:11Z

Replaced content with 'Category: For Deletion'

← Older revision		Revision as of 12:20, 20 July 2012
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	~~=== The Generalized Gamma Distribution ===~~		[[Category: For Deletion]]

	Compared to the other distributions previously discussed, the generalized gamma distribution is not as frequently used for modeling life data; however, it has the the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters. This offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters <span class="texhtml">μ</span> , <span class="texhtml">σ</span> and <span class="texhtml">λ</span> . The ''pdf ''of the distribution is given by,

	~~::<math>~~
	~~f(x)=\begin{cases}~~
	~~\frac{\|\lambda\|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot~~
	~~{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, & \text{if} \ \lambda \ne 0 \\~~

	~~\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \ \lambda =0~~
	~~\end{cases}~~
	~~</math>~~

	~~<br>where <span class="texhtml">Γ(''x'')</span> is the gamma function, defined by: <br>~~

	::~~<math>\Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds</math>~~

	~~<br>~~

	~~This distribution behaves as do other distributions based on the values of the parameters.~~ For example, if <span class="texhtml">λ = 1</span>, then the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

	~~The Generalized Gamma distribution and its characteristics are presented in [[The Generalized Gamma Distribution~~]].

	~~<br>~~

Lisa Hacker: /* The Generalized Gamma Distribution */

2012-03-30T21:44:18Z

The Generalized Gamma Distribution

← Older revision		Revision as of 21:44, 30 March 2012
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	This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, then the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.		This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, then the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

	The Generalized Gamma distribution and its characteristics are presented in ~~detail in the chapter~~ [[The Generalized Gamma Distribution]]		The Generalized Gamma distribution and its characteristics are presented in [[The Generalized Gamma Distribution]].

	<br>		<br>

Kate Racaza at 16:15, 12 March 2012

2012-03-12T16:15:19Z

← Older revision		Revision as of 16:15, 12 March 2012
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	This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, then the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.		This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, then the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

	The Generalized Gamma distribution and its characteristics are presented in ~~more~~ detail in the chapter [[The Generalized Gamma Distribution]]		The Generalized Gamma distribution and its characteristics are presented in detail in the chapter [[The Generalized Gamma Distribution]]

	<br>		<br>

Kate Racaza at 16:13, 12 March 2012

2012-03-12T16:13:20Z

← Older revision		Revision as of 16:13, 12 March 2012
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	=== The Generalized Gamma Distribution ===		=== The Generalized Gamma Distribution ===
	~~While~~ not as frequently used for modeling life data ~~as the distributions discussed previously~~, the ~~generalized gamma distribution does have~~ the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters ~~and also~~ offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters <span class="texhtml">μ</span> , <~~math~~>~~\sigma~~</~~math~~> and <span class="texhtml">λ</span> . The pdf of the distribution is given by,
			Compared to the other distributions previously discussed, the generalized gamma distribution is not as frequently used for modeling life data; however, it has the the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters. This offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters <span class="texhtml">μ</span> , <span class="texhtml">σ</span> and <span class="texhtml">λ</span> . The ''pdf ''of the distribution is given by,

	::<math>		::<math>
Line 10:		Line 11:
	\end{cases}		\end{cases}
	</math>		</math>
	<br>
	where <span class="texhtml">Γ(''x'')</span> is the gamma function, defined by:		<br>where <span class="texhtml">Γ(''x'')</span> is the gamma function, defined by: <br>
	<br>
	::<math>\Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds</math>		::<math>\Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds</math>

	<br>		<br>

	This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, the distribution is identical to the exponential distribution and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.		This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, then the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

	The Generalized Gamma distribution and its characteristics are presented in more detail in ~~Chapter~~ [[The Generalized Gamma Distribution]]		The Generalized Gamma distribution and its characteristics are presented in more detail in the chapter [[The Generalized Gamma Distribution]]

	<br>		<br>

Harry Guo: /* The Generalized Gamma Distribution */

2012-02-11T00:00:01Z

The Generalized Gamma Distribution

← Older revision		Revision as of 00:00, 11 February 2012
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	This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, the distribution is identical to the exponential distribution and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.		This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, the distribution is identical to the exponential distribution and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

	The Generalized Gamma distribution and its characteristics are presented in more detail in [[The Generalized Gamma Distribution ~~\| Chapter 12~~]]		The Generalized Gamma distribution and its characteristics are presented in more detail in Chapter [[The Generalized Gamma Distribution]]

	<br>		<br>

Harry Guo: /* The Generalized Gamma Distribution */

2012-02-03T23:01:04Z

The Generalized Gamma Distribution

← Older revision		Revision as of 23:01, 3 February 2012
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	f(x)=\begin{cases}		f(x)=\begin{cases}
	\frac{\|\lambda\|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot		\frac{\|\lambda\|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot
	{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, &\\\text{if} \lambda \ne 0 \\		{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, & \text{if} \ \lambda \ne 0 \\

	\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \lambda =0		\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \ \lambda =0
	\end{cases}		\end{cases}
	</math>		</math>

Harry Guo: /* The Generalized Gamma Distribution */

2012-02-03T22:57:40Z

The Generalized Gamma Distribution

← Older revision		Revision as of 22:57, 3 February 2012
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	f(x)=\begin{cases}		f(x)=\begin{cases}
	\frac{\|\lambda\|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot		\frac{\|\lambda\|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot
	{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, &\text{if} \lambda \ne 0 \\		{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, &\\\text{if} \lambda \ne 0 \\

	\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \lambda =0		\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \lambda =0

Harry Guo: /* The Generalized Gamma Distribution */

2012-02-03T22:57:21Z

The Generalized Gamma Distribution

← Older revision		Revision as of 22:57, 3 February 2012
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	This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, the distribution is identical to the exponential distribution and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.		This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, the distribution is identical to the exponential distribution and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.

			The Generalized Gamma distribution and its characteristics are presented in more detail in [[The Generalized Gamma Distribution \| Chapter 12]]

	<br>		<br>

Nicolette Young: Created page with '=== The Generalized Gamma Distribution === While not as frequently used for modeling life data as the distributions discussed previously, the generalized gamma distribution does…'

2012-01-03T23:10:30Z

Created page with '=== The Generalized Gamma Distribution === While not as frequently used for modeling life data as the distributions discussed previously, the generalized gamma distribution does…'

New page

=== The Generalized Gamma Distribution ===
While not as frequently used for modeling life data as the distributions discussed previously, the generalized gamma distribution does have the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters and also offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters μ , <math>\sigma</math> and λ . The pdf of the distribution is given by,

::<math>
f(x)=\begin{cases}
\frac{|\lambda|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot
{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, &\text{if} \lambda \ne 0 \\

\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \lambda =0
\end{cases}
</math>
 
where Γ(''x'') is the gamma function, defined by:
 
::<math>\Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds</math>
 

This distribution behaves as do other distributions based on the values of the parameters. For example, if λ = 1, the distribution is identical to the Weibull distribution. If both λ = 1 and σ = 1, the distribution is identical to the exponential distribution and for λ = 0, it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.