Kumaraswamy distribution

Characterization

Probability density function

The probability density function of the Kumaraswamy distribution without considering any inflation is

: f(x; a,b) = a b x^{a-1}{ (1-x^a)}^{b-1}, \ \ \mbox{where} \ \ x \in (0,1),

and where a and b are non-negative shape parameters.

Cumulative distribution function

The cumulative distribution function is

:F(x; a,b)=\int_{0}^{x}f(\xi; a,b) d\xi = 1-(1-x^a)^b.\

Quantile function

The inverse cumulative distribution function (quantile function) is

:F^{-1}(y; a,b)=(1-(1-y)^\frac{1}{b})^\frac{1}{a}. \

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}}. ,!

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 and Γ(.) denotes the Gamma 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.

The Shannon entropy (in nats) of the distribution is:

:H=\left(1!-!\tfrac{1}{a}\right)+\left(1!-!\tfrac{1}{b}\right)H_b-\ln(ab)

where H_i 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 \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 Kumaraswamy distributions by considering random variables of the form Y^{1/\gamma}{\alpha,\beta}, with \gamma0 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 re-obtain 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), (Uniform distribution)
• If X \sim U(0,1) , then {\left( 1-X^{\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 -\log(X) \sim \textrm{Exponential}(a),
• If X \sim \textrm{Kumaraswamy}(1,b), then -\log(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
• If X \sim \textrm{Kumaraswamy}(a,b), then \left{\frac{1}{1-X} \right} \sim \textrm{MK}(a, b),, the Modified Kumaraswamy distribution.

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.

References

References

1. Kumaraswamy, P.. (1980). "A generalized probability density function for double-bounded random processes". Journal of Hydrology.
2. (1996). "Estimation of reservoir yield and storage distribution using moments analysis". Journal of Hydrology.
3. Lemonte, Artur J.. (2011). "Improved point estimation for the Kumaraswamy distribution". Journal of Statistical Computation and Simulation.
4. (2019). "Inflated Kumaraswamy distributions". Anais da Academia Brasileira de Ciências.
5. (2013). "Handbook of Differential Entropy". Chapman and Hall/CRC.
6. Jones, M.C.. (2009). "Kumaraswamy's distribution: A beta-type distribution with some tractability advantages". Statistical Methodology.