# Kumaraswamy distribution

In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous probability distributions defined on the interval (0,1). 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[1] for variables that are lower and upper bounded.

Parameters Probability density function Cumulative distribution function ${\displaystyle a>0\,}$ (real)${\displaystyle b>0\,}$ (real) ${\displaystyle x\in (0,1)\,}$ ${\displaystyle abx^{a-1}(1-x^{a})^{b-1}\,}$ ${\displaystyle 1-(1-x^{a})^{b}}$ ${\displaystyle {\frac {b\Gamma (1+{\tfrac {1}{a}})\Gamma (b)}{\Gamma (1+{\tfrac {1}{a}}+b)}}\,}$ ${\displaystyle \left(1-2^{-1/b}\right)^{1/a}}$ ${\displaystyle \left({\frac {a-1}{ab-1}}\right)^{1/a}}$ for ${\displaystyle a\geq 1,b\geq 1,(a,b)\neq (1,1)}$ (complicated-see text) (complicated-see text) (complicated-see text) ${\displaystyle \left(1\!-\!{\tfrac {1}{b}}\right)+\left(1\!-\!{\tfrac {1}{a}}\right)H_{b}-\ln(ab)}$

## Characterization

### Probability density function

The probability density function of the Kumaraswamy distribution is

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

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

### Cumulative distribution function

${\displaystyle F(x;a,b)=\int _{0}^{x}f(\xi ;a,b)d\xi =1-(1-x^{a})^{b}.\ }$

### 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:

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

## Properties

The raw moments of the Kumaraswamy distribution are given by[2][3]:

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

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:

${\displaystyle \sigma ^{2}=m_{2}-m_{1}^{2}.}$

The Shannon entropy (in nats) of the distribution is:[4]

${\displaystyle H=\left(1\!-\!{\tfrac {1}{b}}\right)+\left(1\!-\!{\tfrac {1}{a}}\right)H_{b}-\ln(ab)}$

where ${\displaystyle H_{i}}$ is the harmonic number function.

## Relation to the Beta distribution

The Kumaraswamy distribution is closely related to Beta distribution[5]. 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 ${\displaystyle \alpha =1}$ and ${\displaystyle \beta =b}$. One has the following relation between Xa,b and Y1,b.

${\displaystyle X_{a,b}=Y_{1,b}^{1/a},}$

with equality in distribution.

${\displaystyle \operatorname {P} \{X_{a,b}\leq x\}=\int _{0}^{x}abt^{a-1}(1-t^{a})^{b-1}dt=\int _{0}^{x^{a}}b(1-t)^{b-1}dt=\operatorname {P} \{Y_{1,b}\leq x^{a}\}=\operatorname {P} \{Y_{1,b}^{1/a}\leq x\}.}$

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

${\displaystyle m_{n}={\frac {\Gamma (\alpha +\beta )\Gamma (\alpha +n/\gamma )}{\Gamma (\alpha )\Gamma (\alpha +\beta +n/\gamma )}}.}$

Note that we can re-obtain the original moments setting ${\displaystyle \alpha =1}$, ${\displaystyle \beta =b}$ and ${\displaystyle \gamma =a}$. However, in general, the cumulative distribution function does not have a closed form solution.

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

## 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[6].

## References

1. Kumaraswamy, P. (1980). "A generalized probability density function for double-bounded random processes". Journal of Hydrology. 46 (1–2): 79–88. doi:10.1016/0022-1694(80)90036-0. ISSN 0022-1694.
2. Lemonte, Artur J. (2011). "Improved point estimation for the Kumaraswamy distribution". Journal of Statistical Computation and Simulation. 81 (12): 1971–1982. doi:10.1080/00949655.2010.511621. ISSN 0094-9655.
3. CRIBARI-NETO, FRANCISCO; SANTOS, JÉSSICA (2019). "Inflated Kumaraswamy distributions". Anais da Academia Brasileira de Ciências. 91 (2). doi:10.1590/0001-3765201920180955. ISSN 1678-2690.
4. Michalowicz, Joseph Victor; Nichols, Jonathan M.; Bucholtz, Frank (2013). Handbook of Differential Entropy. Chapman and Hall/CRC. p. 100. ISBN 9781466583177.
5. Jones, M.C. (2009). "Kumaraswamy's distribution: A beta-type distribution with some tractability advantages". Statistical Methodology. 6 (1): 70–81. doi:10.1016/j.stamet.2008.04.001. ISSN 1572-3127.
6. Fletcher, S.G.; Ponnambalam, K. (1996). "Estimation of reservoir yield and storage distribution using moments analysis". Journal of Hydrology. 182 (1–4): 259–275. doi:10.1016/0022-1694(95)02946-x. ISSN 0022-1694.