ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing various statistical quantities related to the GenGamma distribution. More...
Data Types | |
type | distGenGamma_type |
This is the derived type for signifying distributions that are of type GenGamma as defined in the description of pm_distGenGamma. More... | |
interface | getGenGammaCDF |
Generate and return the Cumulative Distribution Function (CDF) of the Generalized Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More... | |
interface | getGenGammaLogPDF |
Generate and return the natural logarithm of the Probability Density Function (PDF) of the GenGamma distribution. More... | |
interface | getGenGammaLogPDFNF |
Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the GenGamma distribution. More... | |
interface | setGenGammaCDF |
Return the Cumulative Distribution Function (CDF) of the Generalized Gamma distribution for an input x within the support of the distribution \(x \in (0,+\infty)\). More... | |
interface | setGenGammaLogPDF |
Return the natural logarithm of the Probability Density Function (PDF) of the GenGamma distribution. More... | |
interface | setGenGammaRand |
Return a scalar or array of arbitrary rank of GenGamma-distributed random values with the specified shape and scale parameters \((\kappa, \omega, \sigma)\) of the Generalized Gamma distribution corresponding to the procedure arguments (kappa, omega, sigma) . More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distGenGamma" |
This module contains classes and procedures for computing various statistical quantities related to the GenGamma distribution.
Specifically, this module contains routines for computing the following quantities of the GenGamma distribution:
A variable \(X\) is said to be Generalized Gamma (GenGamma) distributed if its PDF with the scale \(0 < \sigma < +\infty\), shape \(0 < \omega < +\infty\), and shape \(0 < \kappa < +\infty\) parameters is described by the following equation,
\begin{equation} \large \pi(x | \kappa, \omega, \sigma) = \frac{1}{\sigma \omega \Gamma(\kappa)} ~ \bigg( \frac{x}{\sigma} \bigg)^{\frac{\kappa}{\omega} - 1} \exp\Bigg( -\bigg(\frac{x}{\sigma}\bigg)^{\frac{1}{\omega}} \Bigg) ~,~ 0 < x < \infty \end{equation}
where \(\eta = \frac{1}{\sigma \omega \Gamma(\kappa)}\) is the normalization factor of the PDF.
When \(\sigma = 1\), the GenGamma PDF simplifies to the form,
\begin{equation} \large \pi(x) = \frac{1}{\omega \Gamma(\kappa)} ~ x^{\frac{\kappa}{\omega} - 1} \exp\Bigg( -x^{\frac{1}{\omega}} \Bigg) ~,~ 0 < x < \infty \end{equation}
If \((\sigma, \omega) = (1, 1)\), the GenGamma PDF further simplifies to the form,
\begin{equation} \large \pi(x) = \frac{1}{\Gamma(\kappa)} ~ x^{\kappa - 1} \exp(-x) ~,~ 0 < x < \infty \end{equation}
Setting the shape parameter to \(\kappa = 1\) further simplifies the PDF to the Exponential distribution PDF with the scale parameter \(\sigma = 1\),
\begin{equation} \large \pi(x) = \exp(x) ~,~ 0 < x < \infty \end{equation}
The CDF of the Generalized Gamma distribution over a strictly-positive support \(x \in (0, +\infty)\) with the three (shape, shape, scale) parameters \((\kappa > 0, \omega > 0, \sigma > 0)\) is defined by the regularized Lower Incomplete Gamma function as,
\begin{eqnarray} \large \mathrm{CDF}(x | \kappa, \omega, \sigma) & = & P\bigg(\kappa, \big(\frac{x}{\sigma}\big)^{\frac{1}{\omega}} \bigg) \\ & = & \frac{1}{\Gamma(\kappa)} \int_0^{\big(\frac{x}{\sigma}\big)^{\frac{1}{\omega}}} ~ t^{\kappa - 1}{\mathrm e}^{-t} ~ dt ~, \end{eqnarray}
where \(\Gamma(\kappa)\) represents the Gamma function.
The distribution mean is given by,
\begin{equation} \large \overline{x} = \frac{\Gamma\left(\kappa + \omega\right)}{\Gamma(\kappa)} \sigma ~. \end{equation}
The distribution mode is given by,
\begin{equation} \large \widehat{x} = \begin{cases} \sigma \left( \kappa - \omega \right)^\omega ~~~ , ~~~ \omega < \kappa ~, \nonumber \\ 0 ~~~ , ~~~ \kappa \leq \omega ~. \end{cases} \end{equation}
The distribution variance is given by,
\begin{equation} \large \mathrm{VAR}(x) = \sigma^2 \left[ \frac{\Gamma(\kappa + 2\omega)}{\Gamma(\kappa)} - \left( \frac{\Gamma(\kappa + \omega)}{\Gamma(\kappa)} \right)^2 \right] ~. \end{equation}
Final Remarks ⛓
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character(*, SK), parameter pm_distGenGamma::MODULE_NAME = "@pm_distGenGamma" |
Definition at line 137 of file pm_distGenGamma.F90.