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 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" |
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:
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.
Final Remarks ⛓
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character(*, SK), parameter pm_distExpGamma::MODULE_NAME = "@pm_distExpGamma" |
Definition at line 97 of file pm_distExpGamma.F90.