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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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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 ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
| character(*, SK), parameter pm_distGenGamma::MODULE_NAME = "@pm_distGenGamma" |
Definition at line 137 of file pm_distGenGamma.F90.
1.9.3