pm_distKolm Module Reference

This module contains classes and procedures for computing various statistical quantities related to the Kolmogorov distribution. More...

Data Types
type	distKolm_type
	This is the derived type for signifying distributions that are of type Kolmogorov as defined in the description of pm_distKolm. More...
interface	getKolmCDF
	Generate and return the Cumulative Distribution Function (CDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More...
interface	getKolmPDF
	Generate and return the Probability Density Function (PDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More...
interface	getKolmQuan
	Generate and return a scalar (or array of arbitrary rank) of the quantile corresponding to the specified CDF of Kolmogorov distribution. More...
interface	getKolmRand
	Generate and return a scalar (or array of arbitrary rank) of the random value(s) from the Kolmogorov distribution. More...
interface	setKolmCDF
	Return the Cumulative Distribution Function (CDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More...
interface	setKolmPDF
	Return the Probability Density Function (PDF) of the Kolmogorov distribution for an input x within the support of the distribution \(X \in [0, +\infty)\). More...
interface	setKolmQuan
	Return a scalar (or array of arbitrary rank) of the quantile corresponding to the specified CDF of Kolmogorov distribution. More...
interface	setKolmRand
	Return a scalar (or array of arbitrary rank) of the random value(s) from the Kolmogorov distribution. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distKolm"

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the Kolmogorov distribution.

Specifically, this module contains routines for computing the following quantities of the Kolmogorov 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 Kolmogorov distribution is the distribution of the random variable

\begin{equation} \large X = \sup_{t\in [0,1]}|B(t)| ~, \end{equation}

where \(B(t)\) is the Brownian bridge.
The cumulative distribution function (CDF) of \(K\) over the non-negative support \(X \in [0, +\infty)\) is given by,

\begin{equation} \large \ms{CDF}(X\leq x) = 1 - 2\sum_{k=1}^{\infty}(-1)^{k-1}e^{-2k^{2}x^{2}} = {\frac{\sqrt{2\pi}}{x}}\sum_{k=1}^{\infty}e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})} ~, \end{equation}

which can also be expressed by the Jacobi theta function \(\vartheta_{01}(z=0;\tau =2ix^{2}/\pi)\).
The distribution is named after Andrey Kolmogorov.

The corresponding PDF \(\pi(\cdot)\) of the Kolmogorov distribution can be obtained by taking the derivative of the CDF of the distribution with respect to \(x\),

\begin{eqnarray} \large \pi(x) &=& 8x \sum_{k=1}^{\infty}(-1)^{k-1} k^2 e^{-2k^{2}x^{2}} \nonumber \\ &=& {\frac{\sqrt{2\pi}}{x^4}}\sum_{k=1}^{\infty} \left[(2k - 1)^2\pi^2/4 - x^2\right] e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})} \nonumber \\ &=& {\frac{2\sqrt{2\pi}}{x^2}}\sum_{k=1}^{\infty} \left[\frac{(2k - 1)^2\pi^2}{8x^2} - \frac{1}{2}\right] e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})} ~, \end{eqnarray}

where the symbol \(\pi\) on the righthand side represents the mathematical number \(\pi = 3.1415\ldots\).

Quantile Function

There is no close form expression for the inverse CDF of the Kolmogorov distribution.
However, root finding methods can be used to refine an initial guess toward an acceptable answer.

Random Number Generation

In the most naive scenario, the quantile function can be used for random number generation.

See also

pm_distUnif
pm_distanceKolm

Test:

test_pm_distKolm

Todo:

Normal Priority: Two additional interfaces for computing the quantiles and random values of Kolmogorov Distribution must be added.
The methodology employed for the [Beta distribution](@ pm_distBeta) might be useful here.

Final Remarks ⛓

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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.

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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_distKolm::MODULE_NAME = "@pm_distKolm"

Definition at line 90 of file pm_distKolm.F90.