Kumaraswamy distribution

From Wikipedia, the free encyclopedia

For other uses, see Kumaraswamy (disambiguation).
Kumaraswamy
Probability density function
Probability density function
Cumulative distribution function
Cumulative density function
Parameters a>0\, (real)
b>0\, (real)
Support x \in [0,1]\,
Probability density function (pdf) abx^{a-1}(1-x^a)^{b-1}\,
Cumulative distribution function (cdf) [1-(1-x^a)^b]\,
Mean \frac{b\Gamma(1+1/a)\Gamma(b)}{\Gamma(1+1/a+b)}\,
Median \left(1-\left(\frac{1}{2}\right)^{1/b}\right)^{1/a}
Mode \left(\frac{a-1}{ab-1}\right)^{1/a}
Variance (complicated-see text)
Skewness (complicated-see text)
Excess kurtosis (complicated-see text)
Entropy
Moment-generating function (mgf)
Characteristic function

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 as versatile as the Beta distribution, but much simpler to use especially in simulation studies due to the simple closed form of both its pdf and cdf. The distribution form was originally presented by Poondi Kumaraswamy for variables that are lower and upper bounded.

Contents

The probability density function of the Kumaraswamy distribution is

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

The cumulative distribution function is therefore

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

In its simplest form, the distribution has a range of [0,1]. In a more general form, we may replace the normalized variable x with the unshifted and unscaled variable z where:

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

The distribution is sometimes combined with a "pike probability" or a Dirac delta function, e.g.:

g(x|a,b) = F_0\delta(x)+(1-F_0)a b x^{a-1}{ (1-x^a)}^{b-1} \,\!

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,n/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:

\sigma^2=m_2-m_1^2

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, 1996).

  • Kumaraswamy, P. (1980). "A generalized probability density function for double-bounded random processes". Journal of Hydrology 46: 79–88. 
  • Fletcher, S.G., and Ponnambalam, K. (1996). "Estimation of reservoir yield and storage distribution using moments analysis". Journal of Hydrology 182: 259–275. 
Image:Bvn-small.png Probability distributions [ view • talk • edit ]
Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular
Retrieved from "http://en.wikipedia.org../../../k/u/m/Kumaraswamy_distribution.html"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.