pm_distGamma Module Reference

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

Data Types
type	distGamma_type
	This is the derived type for signifying distributions that are of type Gamma as defined in the description of pm_distGamma. More...
interface	getGammaCDF
	Generate and return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More...
interface	getGammaLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More...
interface	getGammaLogPDFNF
	Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the Gamma distribution for an input parameter set \((\kappa,\sigma)\). More...
interface	setGammaCDF
	Return the Cumulative Distribution Function (CDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More...
interface	setGammaLogPDF
	Return the natural logarithm of the Probability Density Function (PDF) of the Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More...
interface	setGammaRand
	Return a scalar or array of arbitrary rank of Gamma-distributed random values with the specified shape and scale parameters \((\kappa, \sigma)\) of the Gamma distribution corresponding to the procedure arguments (kappa, sigma). More...

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

Detailed Description

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

Specifically, this module contains routines for computing the following quantities of the Gamma 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

The PDF of the Gamma distribution over a strictly-positive support \(x \in (0, +\infty)\) is defined with the two (shape, scale) parameters \((\kappa > 0, \sigma > 0)\) as,

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

where \(\eta = \frac{1}{\sigma\Gamma(\kappa)}\) is the normalization factor of the distribution with \(\Gamma(\kappa)\) representing the Gamma function whose natural logarithm is returned by the Fortran intrinsic log_gamma().
Note that \(\lim_{x\to0} \pi(x | 0 < \kappa < 1, \sigma) \to +\infty\) \(\).
However, this divergence is integrable and the Gamma PDF is properly normalized.

The CDF of the Gamma 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(\kappa, \frac{x}{\sigma}) \\ & = & \frac{1}{\Gamma(\kappa)} \int_0^{\frac{x}{\sigma}} ~ t^{\kappa - 1}{\mathrm e}^{-t} ~ dt ~, \end{eqnarray}

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

Note

The Cumulative Distribution Function (CDF) of the Gamma distribution for a triple \((x, \kappa, \sigma)\) is directly returned by calling getGammaIncLow(sigma * x, kappa).

See also

Press et al., 1992, Numerical Recipes.
Marsaglia and Tsang, 2000, A simple method for generating gamma variables
Intel Math Kernel Library
Blog Post by Sukhbinder Singh
Blog Post by Yutaka Masuda
cdflib

Test:

test_pm_distGamma

Final Remarks ⛓

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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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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_distGamma::MODULE_NAME = "@pm_distGamma"

Definition at line 83 of file pm_distGamma.F90.