Kumaraswamy distribution


In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous [probability distribution]s defined on the interval. It is similar to the beta distribution, but much simpler to use especially in simulation studies since its probability density function, cumulative distribution function and quantile functions can be expressed in closed form. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded with a zero-inflation. In this first article of the distribution, the natural lower bound of zero for rainfall was modelled using a discrete probability, as rainfall in many places, especially in tropics, has significant nonzero probability. This discrete probability is now called zero-inflation. This was extended to inflations at both extremes in the work of Fletcher and Ponnambalam. A good example for inflations at extremes are the probabilities of full and empty reservoirs and are important for reservoir design.

Characterization

Probability density function

The probability density function of the Kumaraswamy distribution without considering any inflation is
and where a and b are non-negative shape parameters.

Cumulative distribution function

The cumulative distribution function is

Quantile function

The inverse cumulative distribution function is

Generalizing to arbitrary interval support

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

Properties

The raw moments of the Kumaraswamy distribution are given by:
where B is the Beta function and Γ denotes the Gamma function. The variance, skewness, and excess kurtosis can be calculated from these raw moments. For example, the variance is:
The Shannon entropy of the distribution is:
where is the harmonic number function.

Relation to the Beta distribution

The Kumaraswamy 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 and.
One has the following relation between Xa,b and Y1,b.
with equality in distribution.
One may introduce generalised Kumaraswamy distributions by considering random variables of the form
, with and where
denotes a Beta distributed random variable with parameters and.
The raw moments of this generalized Kumaraswamy distribution are given by:
Note that we can re-obtain the original moments setting, and.
However, in general, the cumulative distribution function does not have a closed form solution.

Related distributions

Example

An example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity z whose upper bound is zmax and lower bound is 0, which is also a natural example for having two inflations as many reservoirs have nonzero probabilities for both empty and full reservoir states.