pm_distExpGamma Module Reference

This module contains classes and procedures for computing various statistical quantities related to the ExpGamma distribution. More...

Data Types
type	distExpGamma_type
	This is the derived type for signifying distributions that are of type ExpGamma as defined in the description of pm_distExpGamma. More...
interface	getExpGammaCDF
	Generate and return the Cumulative Distribution Function (CDF) of the ExpGamma distribution for an input x within the support of the distribution \(x \in (-\infty,+\infty)\). More...
interface	getExpGammaLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the ExpGamma distribution. More...
interface	getExpGammaLogPDFNF
	Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the ExpGamma distribution. More...
interface	setExpGammaCDF
	Return the Cumulative Distribution Function (CDF) of the ExpGamma distribution for an input x within the support of the distribution \(x \in (-\infty,+\infty)\). More...
interface	setExpGammaLogPDF
	Return the natural logarithm of the Probability Density Function (PDF) of the ExpGamma distribution. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distExpGamma"

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the ExpGamma distribution.

Specifically, this module contains routines for computing the following quantities of the ExpGamma distribution:

1. the Probability Density Function (PDF)
2. the Cumulative Distribution Function (CDF)
3. the Random Number Generation from the distribution (RNG)
4. the Inverse Cumulative Distribution Function (ICDF) or the Quantile Function

A variable \(X\) is said to be ExpGamma-distributed if its PDF with location \(0 < \log(\sigma) < +\infty\), scale (inverse rate) \(> 0\), and shape \(\kappa > 0\) parameters is described by the following equation,

\begin{equation} \large \pi(x | \kappa, \log(\sigma)) = \frac{1}{\Gamma(\kappa)} ~ \exp\Bigg( \kappa\bigg(x - \log(\sigma)\bigg) - \exp\bigg(x - \log(\sigma)\bigg) \Bigg) ~,~ -\infty < x < \infty \end{equation}

where \(\eta = \frac{1}{\Gamma(\kappa)}\) is the normalization factor of the PDF.

When \(\sigma = 1\), the ExpGamma PDF simplifies to the form,

\begin{equation} \large \pi(x) = \frac{1}{\Gamma(\kappa)} ~ \exp\Bigg(\kappa x - \exp(x) \Bigg) ~,~ -\infty < x < \infty \end{equation}

Setting the shape parameter to \(\kappa = 1\) further simplifies the PDF to the form,

\begin{equation} \large \pi(x) = \exp\Bigg(x - \exp(x)\Bigg) ~,~ -\infty < x < \infty \end{equation}

The parameter \(\log(\sigma)\) determines the horizontal location of the mode of this unimodal ExpGamma PDF.
The ExpGamma distribution is mathematically defined to be the distribution of \(\log(X)\) where \(X\) follows a Gamma distribution.

The CDF of the ExpGamma distribution over a strictly-positive support \(x \in (0, +\infty)\) with the two (shape, scale) parameters \((\kappa > 0, \sigma > 0)\) is defined by the regularized Lower Incomplete Gamma function as,

\begin{eqnarray} \large \mathrm{CDF}(x | \kappa, \sigma) & = & P\big(\kappa, x - \log(\sigma)\big) \\ & = & \frac{1}{\Gamma(\kappa)} \int_0^{\exp(x - \log(\sigma))} ~ t^{\kappa - 1}{\mathrm e}^{-t} ~ dt ~, \end{eqnarray}

where \(\Gamma(\kappa)\) represents the Gamma function.

Warning

The ExpGamma distribution is also frequently mistakenly called LogGamma , which is an entirely different distribution.

Note

The relationship between the ExpGamma and Gamma distributions is similar to that of the Normal and LogNormal distributions.
In other words, a more consistent naming for the ExpGamma and Gamma distributions could have been Gamma and LogGamma respectively, similar to Normal and LogNormal distributions.

See also

pm_distGamma
pm_distExpGamma
pm_distGenGamma
pm_distGenExpGamma

Test:

test_pm_distExpGamma

Final Remarks ⛓

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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Variable Documentation

MODULE_NAME

character(*, SK), parameter pm_distExpGamma::MODULE_NAME = "@pm_distExpGamma"

Definition at line 97 of file pm_distExpGamma.F90.