ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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pm_distPoweto Module Reference

This module contains classes and procedures for computing various statistical quantities related to the (Truncated) Power/Pareto distribution (hence the name Poweto). More...

Data Types

type  distPoweto_type
 This is the derived type for signifying distributions that are of type Poweto as defined in the description of pm_distPoweto. More...
 
interface  getPowetoLogCDF
 Generate and return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Truncated) Poweto distribution for an input logx within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\).
More...
 
interface  getPowetoLogCDFNF
 Generate and return the natural logarithm of the normalization factor of the Cumulative Distribution Function (CDF) of the (Truncated) Poweto distribution for the input parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
 
interface  getPowetoLogPDF
 Generate and return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) Poweto distribution for an input logx within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\).
More...
 
interface  getPowetoLogPDFNF
 Generate and return the natural logarithm of the normalization factor of the Probability Density Function (PDF) of the (Truncated) Poweto distribution for the input parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
 
interface  getPowetoLogQuan
 Generate and return the natural logarithm of the Inverse Cumulative Distribution (Quantile) Function of the (Truncated) Poweto distribution for an input logCDF within the support of the distribution \((\alpha, x_\mathrm{min}, x_\mathrm{max})\).
More...
 
interface  getPowetoLogRand
 Generate and return a scalar (or array of arbitrary rank) of the natural logarithm(s) of random value(s) from the (Truncated) Poweto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\).
More...
 
interface  setPowetoLogCDF
 Return the natural logarithm of the Cumulative Distribution Function (CDF) of the (Truncated) Poweto distribution for an input logx within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\).
More...
 
interface  setPowetoLogPDF
 Return the natural logarithm of the Probability Density Function (PDF) of the (Truncated) Poweto distribution for an input logx within the support of the distribution \(x \in [0 < x_\mathrm{min}, x_\mathrm{max} < +\infty]\).
More...
 
interface  setPowetoLogQuan
 Return the natural logarithm of the Inverse Cumulative Distribution (Quantile) Function of the (Truncated) Poweto distribution for an input logCDF within the support of the distribution \((\alpha, x_\mathrm{min}, x_\mathrm{max})\).
More...
 
interface  setPowetoLogRand
 Return a scalar (or array of arbitrary rank) of the natural logarithm(s) of random value(s) from the (Truncated) Poweto distribution with parameters \((\alpha, x_\mathrm{min}, x_\mathrm{max})\). More...
 

Variables

character(*, SK), parameter MODULE_NAME = "@pm_distPoweto"
 

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the (Truncated) Power/Pareto distribution (hence the name Poweto).

Specifically, this module contains routines for computing the following quantities of the (Truncated) Power/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 (Truncated) Poweto distribution is a generalization of the (Truncated) Pareto and Power distributions.

The PDF of the (Truncated) Poweto 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 \(\mathbf{-\infty < \alpha < +\infty}\) 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.

When \(\alpha = 0\), the (Truncated) Poweto distribution simplifies to the Power-law distribution with exponent \(-1\).
When \(0 < \alpha < +\infty\), the (Truncated) Poweto distribution simplifies to (Truncated) Power distribution.
When \(-\infty < \alpha < 0\), the (Truncated) Poweto distribution simplifies to (Truncated) Pareto distribution.

See also
pm_distUnif
pm_distPower
pm_distPareto
pm_distPoweto
pm_distPiwiPoweto
Test:
test_pm_distPoweto
Todo:
Generic interfaces for computing the logarithm of CDF robustly (without numerical rounding) must be added in the future.


Final Remarks


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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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Author:
Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Variable Documentation

◆ MODULE_NAME

character(*, SK), parameter pm_distPoweto::MODULE_NAME = "@pm_distPoweto"

Definition at line 73 of file pm_distPoweto.F90.