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 (Truncated) Pareto distribution. More...
Data Types | |
type | distPareto_type |
This is the derived type for signifying distributions that are of type Pareto as defined in the description of pm_distPareto. More... | |
interface | getParetoLogCDF |
Generate and return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Truncated) Pareto distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More... | |
interface | getParetoLogCDFNF |
Generate and return the natural logarithm of the normalization factor of the CDF of the (Truncated) Pareto distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More... | |
interface | getParetoLogPDF |
Generate and return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) Pareto distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More... | |
interface | getParetoLogPDFNF |
Generate and return the natural logarithm of the normalization factor of the PDF of the (Truncated) Pareto distribution for an input parameter set \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More... | |
interface | getParetoLogQuan |
Generate and return a scalar (or array of arbitrary rank) of the natural logarithm(s) of quantile corresponding to the specified CDF of (Truncated) Pareto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More... | |
interface | getParetoLogRand |
Generate and return a scalar (or array of arbitrary rank) of the natural logarithm(s) of random value(s) from the (Truncated) Pareto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More... | |
interface | setParetoLogCDF |
Return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Truncated) Pareto distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More... | |
interface | setParetoLogPDF |
Return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) Pareto distribution for an input log(x) within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\). More... | |
interface | setParetoLogQuan |
Return a scalar (or array of arbitrary rank) of the natural logarithm(s) of quantile corresponding to the specified CDF of (Truncated) Pareto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More... | |
interface | setParetoLogRand |
Return a scalar (or array of arbitrary rank) of the natural logarithm(s) of random value(s) from the (Truncated) Pareto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distPareto" |
This module contains classes and procedures for computing various statistical quantities related to the (Truncated) Pareto distribution.
Specifically, this module contains routines for computing the following quantities of the (Truncated) Pareto distribution:
The PDF of the (Truncated) Pareto distribution over a strictly-positive support \(x \in [x_\mathrm{min}, x_\mathrm{max}]\) is defined with the three (shape, scale, scale) parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\) as,
\begin{equation} \large \pi(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) ~ x^{\alpha - 1} ~, \end{equation}
where \(\mathbf{-\infty < \alpha < 0}\) and \(\mathbf{0 < x_\mathrm{min} \leq x \leq x_\mathrm{max} < +\infty}\) hold, and \(\eta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the PDF,
\begin{equation} \large \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{\alpha}{x_\mathrm{max}^\alpha - x_\mathrm{min}^\alpha} ~, \end{equation}
When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Pareto distribution simplifies to the Pareto Distribution with PDF,
\begin{equation} \large \lim_{x_\mathrm{max} \rightarrow +\infty} \pi(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{-\alpha}{x_\mathrm{min}^\alpha} x^{\alpha - 1} ~, \end{equation}
The equation for \(\eta(\cdot)\) for the Pareto distribution simplifies to,
\begin{equation} \large \lim_{x_\mathrm{max} \rightarrow +\infty} \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = -\alpha x_\mathrm{min}^{-\alpha} ~,~ 0 < \alpha < +\infty. \end{equation}
The corresponding CDF of the (Truncated) Pareto distribution is given by,
\begin{equation} \large \mathrm{CDF}(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) \bigg[1 - \bigg(\frac{x}{x_\mathrm{min}}\bigg)^\alpha\bigg] ~, \end{equation}
where \(\mathbf{0 < \alpha < +\infty}\) and \(\mathbf{0 < x_\mathrm{min} \leq x \leq x_\mathrm{max} < +\infty}\) hold, and \(\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the CDF,
\begin{eqnarray} \large \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) &=& \frac{x_\mathrm{min}^\alpha}{x_\mathrm{min}^\alpha - x_\mathrm{max}^\alpha} ~, \\ &=& -\frac{x_\mathrm{min}^\alpha}{\alpha} \eta(\alpha, x_\mathrm{min}, x_\mathrm{max}) ~, \end{eqnarray}
where \(\eta(\cdot)\) is the normalization factor of the PDF.
When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Pareto distribution simplifies to the Pareto Distribution with CDF,
\begin{equation} \large \lim_{x_\mathrm{max} \rightarrow +\infty} \mathrm{CDF}(x | \alpha, x_\mathrm{min}, x_\mathrm{max}) = \bigg[1 - \bigg(\frac{x}{x_\mathrm{min}}\bigg)^\alpha\bigg] ~, \end{equation}
with,
\begin{equation} \large \lim_{x_\mathrm{max} \rightarrow +\infty} \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = +1 ~. \end{equation}
The corresponding Inverse CDF of the (Truncated) Pareto distribution is given by,
\begin{equation} \large Q(\mathrm{CDF}(x); \alpha, x_\mathrm{min}, x_\mathrm{max}) \equiv x = x_\mathrm{min} \bigg(1 - \frac{\mathrm{CDF}(x)}{\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})}\bigg)^{\frac{1}{\alpha}} ~, \end{equation}
where \(\mathbf{-\infty < \alpha < 0}\) and \(\mathbf{0 < x_\mathrm{min} \leq Q(\mathrm{CDF}(x); \alpha, x_\mathrm{min}, x_\mathrm{max}) \leq x_\mathrm{max} < +\infty}\) hold, and \(\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the CDF,
\begin{equation} \large \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{x_\mathrm{min}^\alpha}{x_\mathrm{min}^\alpha - x_\mathrm{max}^\alpha} ~. \end{equation}
When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Pareto distribution simplifies to the Pareto Distribution, with,
\begin{equation} \large Q(\mathrm{CDF}(x); \alpha, x_\mathrm{min}, x_\mathrm{max}) \equiv x = x_\mathrm{min} \big(1 - \mathrm{CDF}(x)\big)^{\frac{1}{\alpha}} ~, \end{equation}
Random Number Generation
Assuming that \(U \in [0, 1)\) is a uniformly-distributed random variate, the transformed random variable,
\begin{equation} \large x = x_\mathrm{min} \bigg(1 - \frac{U}{\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})}\bigg)^{\frac{1}{\alpha}} ~, \end{equation}
follows a Truncated Pareto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max}\)) where \(\mathbf{-\infty < \alpha < 0}\) and \(\mathbf{0 < x_\mathrm{min} \leq x \leq x_\mathrm{max} < +\infty}\) hold, and \(\zeta(\alpha, x_\mathrm{min}, x_\mathrm{max})\) is the normalization factor of the CDF,
\begin{equation} \large \zeta(\alpha, x_\mathrm{min}, x_\mathrm{max}) = \frac{x_\mathrm{min}^\alpha}{x_\mathrm{min}^\alpha - x_\mathrm{max}^\alpha} ~, \end{equation}
When \(x_\mathrm{max} \rightarrow +\infty\), the Truncated Pareto distribution simplifies to the Pareto Distribution, with,
\begin{equation} \large x = x_\mathrm{min} \big(1 - U\big)^{\frac{1}{\alpha}} ~, \end{equation}
Final Remarks ⛓
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character(*, SK), parameter pm_distPareto::MODULE_NAME = "@pm_distPareto" |
Definition at line 147 of file pm_distPareto.F90.