World Library  
Flag as Inappropriate
Email this Article

Kumaraswamy distribution

Article Id: WHEBN0002049583
Reproduction Date:

Title: Kumaraswamy distribution  
Author: World Heritage Encyclopedia
Language: English
Subject: Logit-normal distribution, Probability Distribution, Hyper-Erlang distribution, Chernoff's distribution, Extended negative binomial distribution
Publisher: World Heritage Encyclopedia

Kumaraswamy distribution

Probability density function
Probability density function
Cumulative distribution function
Cumulative distribution function
Parameters a>0\, (real)
b>0\, (real)
Support x \in [0,1]\,
pdf abx^{a-1}(1-x^a)^{b-1}\,
CDF [1-(1-x^a)^b]\,
Mean \frac{b\Gamma(1+\tfrac{1}{a})\Gamma(b)}{\Gamma(1+\tfrac{1}{a}+b)}\,
Median \left(1-2^{-1/b}\right)^{1/a}
Mode \left(\frac{a-1}{ab-1}\right)^{1/a} for a\geq 1, b\geq 1, (a,b)\neq (1,1)
Variance (complicated-see text)
Skewness (complicated-see text)
Ex. kurtosis (complicated-see text)

In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous probability distributions defined on the interval [0,1] differing in the values of their two non-negative shape parameters, a and b.

It is similar to the Beta distribution, but much simpler to use especially in simulation studies due to the simple closed form of both its probability density function and cumulative distribution function. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded.


Probability density function

The probability density function of the Kumaraswamy distribution is

f(x; a,b) = a b x^{a-1}{ (1-x^a)}^{b-1}.

Cumulative distribution function

The cumulative distribution function is therefore

F(x; a,b)=1-(1-x^a)^b.\

Generalizing to arbitrary interval support

In its simplest form, the distribution has a support of [0,1]. In a more general form, the normalized variable x is replaced with the unshifted and unscaled variable z where:

x = \frac{z-z_{\text{min}}}{z_{\text{max}}-z_{\text{min}}} , \qquad z_{\text{min}} \le z \le z_{\text{max}}. \,\!


The raw moments of the Kumaraswamy distribution are given by :

m_n = \frac{b\Gamma(1+n/a)\Gamma(b)}{\Gamma(1+b+n/a)} = bB(1+n/a,b)\,

where B is the Beta function. The variance, skewness, and excess kurtosis can be calculated from these raw moments. For example, the variance is:


Relation to the Beta distribution

The Kuramaswamy distribution is closely related to Beta distribution. Assume that Xa,b is a Kumaraswamy distributed random variable with parameters a and b. Then Xa,b is the a-th root of a suitably defined Beta distributed random variable. More formally, Let Y1,b denote a Beta distributed random variable with parameters \alpha=1 and \beta=b. One has the following relation between Xa,b and Y1,b.


with equality in distribution.

\operatorname{P}\{X_{a,b}\le x\}=\int_0^x ab t^{a-1}(1-t^a)^{b-1}dt= \int_0^{x^a} b(1-t)^{b-1}dt=\operatorname{P}\{Y_{1,b}\le x^a\} =\operatorname{P}\{Y^{1/a}_{1,b}\le x\} .

One may introduce generalised Kumaraswamy distributions by considering random variables of the form Y^{1/\gamma}_{\alpha,\beta}, with \gamma>0 and where Y_{\alpha,\beta} denotes a Beta distributed random variable with parameters \alpha and \beta. The raw moments of this generalized Kumaraswamy distribution are given by:

m_n = \frac{\Gamma(\alpha+\beta)\Gamma(\alpha+n/\gamma)}{\Gamma(\alpha)\Gamma(\alpha+\beta+n/\gamma)}.

Note that we can reobtain the original moments setting \alpha=1, \beta=b and \gamma=a. However, in general the cumulative distribution function does not have a closed form solution.

Related distributions

  • If X \sim \textrm{Kumaraswamy}(1,1)\, then X \sim U(0,1)\,
  • If X \sim U(0,1) \, (Uniform distribution (continuous)) then {\left( 1 - {\left( 1-X \right) }^{\tfrac{1}{b}} \right) }^{ \tfrac{1}{a} } \sim \textrm{Kumaraswamy}(a,b)\,
  • If X \sim \textrm{Beta}(1,b) \, (Beta distribution) then X \sim \textrm{Kumaraswamy}(1,b)\,
  • If X \sim \textrm{Beta}(a,1) \, (Beta distribution) then X \sim \textrm{Kumaraswamy}(a,1)\,
  • If X \sim \textrm{Kumaraswamy}(a,1)\, then (1-X) \sim \textrm{Kumaraswamy}(1, a)\,
  • If X \sim \textrm{Kumaraswamy}(1,a)\, then (1-X) \sim \textrm{Kumaraswamy}(a, 1)\,
  • If X \sim \textrm{Kumaraswamy}(a,1)\, then -ln(X) \sim \textrm{Exponential}(a)\,
  • If X \sim \textrm{Kumaraswamy}(1,b)\, then -ln(1-X) \sim \textrm{Exponential}(b)\,
  • If X \sim \textrm{Kumaraswamy}(a,b)\, then X \sim \textrm{GB1}(a, 1, 1, b)\,, the generalized beta distribution of the first kind.


A good example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity zmax whose upper bound is zmax and lower bound is 0 (Fletcher & Ponnambalam, 1996).


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.