pm_distPareto Module Reference

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"

Detailed Description

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:

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 (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}

Remarks

See pm_distPower for the case of \(0 < \alpha < +\infty\).
See pm_distPoweto for the case of \(\alpha = 0\).

See also

pm_distPower
pm_distPareto
pm_distPoweto

Test:

test_pm_distPareto

Final Remarks ⛓

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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_distPareto::MODULE_NAME = "@pm_distPareto"

Definition at line 147 of file pm_distPareto.F90.