Kumaraswamy distribution


 * For other uses, see Kumaraswamy (disambiguation).

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 range
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.:


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

Properties
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:


 * $$\sigma^2=m_2-m_1^2.$$

Relation to the Beta distribution
The Kuramaswamy distribution is closely related to 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.


 * $$X_{a,b}=Y^{1/a}_{1,b},$$

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 Kuramaswamy 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.

Example
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).