Template:Generalized gamma probability density function - Revision history

Lisa Hacker: Redirected page to The Generalized Gamma Distribution

2012-08-09T09:36:38Z

Redirected page to The Generalized Gamma Distribution

← Older revision		Revision as of 09:36, 9 August 2012
Line 1:		Line 1:
	~~===Generalized Gamma Probability Density Function===~~		#REDIRECT [[The_Generalized_Gamma_Distribution]]

	The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters ''k'', <math>\beta</math>, and <math>\theta </math>. The ''pdf'' for this form of the generalized gamma distribution is given by:

	~~::<math>f(t)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\beta -1}}{{e}^{-{{\left( \frac{t}{\theta } \right)}^{\beta }}}}</math>~~

	~~where <math>\theta >0</math> is a scale parameter, <math>\beta >0</math> and <math>k>0</math> are shape parameters and <math>\Gamma (x)</math> is the gamma function of ''x'', which is defined by:~~

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

	With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of ''k'', <math>\beta</math>, and <math>\theta </math> may appear almost identical [[Appendix: Weibull References\|[21]]]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters ''k'', <math>\beta</math>, and <math>\theta </math> [[Appendix: Weibull References\|[21]]~~] where:~~

	~~::<math>\begin{align}~~
	~~& \mu =\ln (\theta )+\frac{1}{\beta }\cdot \ln \left( \frac{1}{{{\lambda }^{2}}} \right) \\~~
	~~& \\~~
	~~& \sigma =\frac{1}{\beta \sqrt{k}} \\~~
	~~& \\~~
	~~& \lambda =\frac{1}{\sqrt{k}} \\~~
	~~\end{align}</math>~~

	~~where <math>-\infty <\mu <\infty ,\text{ }\sigma >0\text{, }0<\lambda .</math>~~

	While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing <math>\lambda </math> to become negative, the ''pdf'' of the reparameterized distribution is given by:


	~~::<math>f(t)=\left\{ \begin{matrix}~~
	\tfrac{\|\lambda \|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma \left( \tfrac{1}{{{\lambda }^{2}}} \right)}\cdot {{e}^{\left[ \tfrac{\lambda \cdot \tfrac{\text{ln}(t)-\mu }{\sigma }+\text{ln}\left( \tfrac{1}{{{\lambda }^{2}}} \right)-{{e}^{\lambda \cdot \tfrac{\text{ln}(t)-\mu }{\sigma }}}}{{{\lambda }^{2}}} \right]}}\text{ if }\lambda \ne 0 \\
	~~\tfrac{1}{t\cdot \sigma \sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}^{2}}}}\text{ if }\lambda =0 \\~~
	~~\end{matrix} \right.</math>~~

Harry Guo: /* Generalized Gamma Probability Density Function */

2012-02-14T20:58:23Z

Generalized Gamma Probability Density Function

← Older revision		Revision as of 20:58, 14 February 2012
Line 22:		Line 22:

	While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing <math>\lambda </math> to become negative, the ''pdf'' of the reparameterized distribution is given by:		While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing <math>\lambda </math> to become negative, the ''pdf'' of the reparameterized distribution is given by:


			::<math>f(t)=\left\{ \begin{matrix}
			\tfrac{\|\lambda \|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma \left( \tfrac{1}{{{\lambda }^{2}}} \right)}\cdot {{e}^{\left[ \tfrac{\lambda \cdot \tfrac{\text{ln}(t)-\mu }{\sigma }+\text{ln}\left( \tfrac{1}{{{\lambda }^{2}}} \right)-{{e}^{\lambda \cdot \tfrac{\text{ln}(t)-\mu }{\sigma }}}}{{{\lambda }^{2}}} \right]}}\text{ if }\lambda \ne 0 \\
			\tfrac{1}{t\cdot \sigma \sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}^{2}}}}\text{ if }\lambda =0 \\
			\end{matrix} \right.</math>

Harry Guo at 19:10, 14 February 2012

2012-02-14T19:10:56Z

← Older revision		Revision as of 19:10, 14 February 2012
Line 11:		Line 11:
	With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of ''k'', <math>\beta</math>, and <math>\theta </math> may appear almost identical [[Appendix: Weibull References\|[21]]]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters ''k'', <math>\beta</math>, and <math>\theta </math> [[Appendix: Weibull References\|[21]]] where:		With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of ''k'', <math>\beta</math>, and <math>\theta </math> may appear almost identical [[Appendix: Weibull References\|[21]]]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters ''k'', <math>\beta</math>, and <math>\theta </math> [[Appendix: Weibull References\|[21]]] where:

			::<math>\begin{align}
			& \mu =\ln (\theta )+\frac{1}{\beta }\cdot \ln \left( \frac{1}{{{\lambda }^{2}}} \right) \\
			& \\
			& \sigma =\frac{1}{\beta \sqrt{k}} \\
			& \\
			& \lambda =\frac{1}{\sqrt{k}} \\
			\end{align}</math>

	where ~~and~~ While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing to become negative, the of the reparameterized distribution is given by:		where <math>-\infty <\mu <\infty ,\text{ }\sigma >0\text{, }0<\lambda .</math>

			While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing <math>\lambda </math> to become negative, the ''pdf'' of the reparameterized distribution is given by:

Harry Guo at 19:06, 14 February 2012

2012-02-14T19:06:30Z

← Older revision		Revision as of 19:06, 14 February 2012
Line 7:		Line 7:
	where <math>\theta >0</math> is a scale parameter, <math>\beta >0</math> and <math>k>0</math> are shape parameters and <math>\Gamma (x)</math> is the gamma function of ''x'', which is defined by:		where <math>\theta >0</math> is a scale parameter, <math>\beta >0</math> and <math>k>0</math> are shape parameters and <math>\Gamma (x)</math> is the gamma function of ''x'', which is defined by:

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

	With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of , , and may appear almost identical [21]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters , , and [21] where:		With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of ''k'', <math>\beta</math>, and <math>\theta </math> may appear almost identical [[Appendix: Weibull References\|[21]]]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters ''k'', <math>\beta</math>, and <math>\theta </math> [[Appendix: Weibull References\|[21]]] where:


	where and While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing to become negative, the of the reparameterized distribution is given by:		where and While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing to become negative, the of the reparameterized distribution is given by:

Harry Guo: /* Generalized Gamma Probability Density Function */

2012-02-14T19:04:07Z

Generalized Gamma Probability Density Function

← Older revision		Revision as of 19:04, 14 February 2012
Line 1:		Line 1:
	===Generalized Gamma Probability Density Function===		===Generalized Gamma Probability Density Function===

	The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters ''k'', <math>\beta<\math>, and <math>\theta<\math>. The ''pdf'' for this form of the generalized gamma distribution is given by:		The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters ''k'', <math>\beta</math>, and <math>\theta </math>. The ''pdf'' for this form of the generalized gamma distribution is given by:

	::<math>f(t)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\beta -1}}{{e}^{-{{\left( \frac{t}{\theta } \right)}^{\beta }}}}</math>		::<math>f(t)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\beta -1}}{{e}^{-{{\left( \frac{t}{\theta } \right)}^{\beta }}}}</math>

Harry Guo: /* Generalized Gamma Probability Density Function */

2012-02-14T19:03:38Z

Generalized Gamma Probability Density Function

← Older revision		Revision as of 19:03, 14 February 2012
Line 1:		Line 1:
	===Generalized Gamma Probability Density Function===		===Generalized Gamma Probability Density Function===

	The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters , , and . The for this form of the generalized gamma distribution is given by:		The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters ''k'', <math>\beta<\math>, and <math>\theta<\math>. The ''pdf'' for this form of the generalized gamma distribution is given by:

			::<math>f(t)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\beta -1}}{{e}^{-{{\left( \frac{t}{\theta } \right)}^{\beta }}}}</math>

	where is a scale parameter, and are shape parameters and is the gamma function of , which is defined by:		where <math>\theta >0</math> is a scale parameter, <math>\beta >0</math> and <math>k>0</math> are shape parameters and <math>\Gamma (x)</math> is the gamma function of ''x'', which is defined by:

Harry Guo at 18:39, 14 February 2012

2012-02-14T18:39:12Z

← Older revision		Revision as of 18:39, 14 February 2012
Line 1:		Line 1:
	====Generalized Gamma Probability Density Function====		===Generalized Gamma Probability Density Function===

	The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters , , and . The for this form of the generalized gamma distribution is given by:		The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters , , and . The for this form of the generalized gamma distribution is given by:

Nicolette Young at 23:06, 16 January 2012

2012-01-16T23:06:26Z

← Older revision		Revision as of 23:06, 16 January 2012
Line 1:		Line 1:
	Generalized Gamma Probability Density Function		====Generalized Gamma Probability Density Function====

	The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters , , and . The for this form of the generalized gamma distribution is given by:		The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters , , and . The for this form of the generalized gamma distribution is given by:

Nicolette Young: /* Generalized Gamma Probability Density Function */

2012-01-16T23:06:12Z

Generalized Gamma Probability Density Function

← Older revision	Revision as of 23:06, 16 January 2012
Line 1:	Line 1:
		Generalized Gamma Probability Density Function

		The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters , , and . The for this form of the generalized gamma distribution is given by:


		where is a scale parameter, and are shape parameters and is the gamma function of , which is defined by:


		With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of , , and may appear almost identical [21]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters , , and [21] where:


		where and While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing to become negative, the of the reparameterized distribution is given by:

Nicolette Young: Blanked the page

2012-01-16T23:05:16Z

Blanked the page

← Older revision		Revision as of 23:05, 16 January 2012
Line 1:		Line 1:
	~~{{the generalized gamma distribution}}~~