Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages

Abstract

A two-parameter family of distributions on $(0,1)$ is explored which has many similarities to the beta distribution and a number of advantages in terms of tractability (it also, of course, has some disadvantages). Kumaraswamy’s distribution has its genesis in terms of uniform order statistics, and has particularly straightforward distribution and quantile functions which do not depend on special functions (and hence afford very easy random variate generation). The distribution might, therefore, have a particular role when a quantile-based approach to statistical modelling is taken, and its tractability has appeal for pedagogical uses. To date, the distribution has seen only limited use and development in the hydrological literature.

Introduction

Despite the many alternatives and generalisations [20], [27], it remains fair to say that the beta distribution provides the premier family of continuous distributions on bounded support (which is taken to be $(0,1)$). The beta distribution, $\mathrm{Beta}(a,b)$, has density

$$g\left(x\right)=\frac{1}{B(a,b)}{x}^{a-1}{(1-x)}^{b-1}\text{,}0<x<1\text{,}$$

where its two shape parameters $a$ and $b$ are positive and $B(\cdot ,\cdot )$ is the beta function. Beta densities are unimodal, uniantimodal, increasing, decreasing or constant depending on the values of $a$ and $b$ relative to 1 and have a host of other attractive properties ([15], Chapter 25). The beta distribution is fairly tractable, but in some ways not fabulously so; in particular, its distribution function is an incomplete beta function ratio and its quantile function the inverse thereof.

In this paper, I take a look at an alternative two-parameter distribution on $(0,1)$ which I will call Kumaraswamy’s distribution, $\text{Kumaraswamy}(\alpha ,\beta )$, where I have denoted its two positive shape parameters $\alpha$ and $\beta$. It has many of the same properties as the beta distribution but has some advantages in terms of tractability. Its density is

$$f\left(x\right)=f(x;\alpha ,\beta )=\alpha \beta x^{\alpha -1}{(1-x^{\alpha })}^{\beta -1}\text{,}0<x<1\text{.}$$

Alert readers might recognise it in some way, especially if they are familiar with the hydrological literature where it dates back to [22]. However, the distribution does not seem to be very familiar to statisticians, has not been investigated systematically in much detail before, nor has its relative interchangeability with the beta distribution been widely appreciated. For example, Kumaraswamy’s densities are also unimodal, uniantimodal, increasing, decreasing or constant depending in the same way as the beta distribution on the values of its parameters. (Boundary behaviour and the main special cases are also common to both beta and Kumaraswamy’s distribution.) And yet the normalising constant in (1.2) is very simple and the corresponding distribution and quantile functions also need no special functions. The latter gives Kumaraswamy’s distribution an advantage if viewed from the quantile modelling perspective popular in some quarters [28], [7]. Some other properties are also more readily available, mathematically, than their counterparts for the beta distribution. Yet the beta distribution also has its particular advantages and I hesitate to claim whether, in the end, the tractability advantages of Kumaraswamy’s distribution will prove to be of immense practical significance in statistics; at the very least, Kumaraswamy’s distribution might find a pedagogical role.

The background and genesis of Kumaraswamy’s distribution are given in Section 2 along with its principal special cases. The basic properties of Kumaraswamy’s distribution are given in Section 3 whose easy reading reflects the tractability of the distribution. A deeper investigation of the skewness and kurtosis properties of the distribution is given in Section 4. Inference by maximum likelihood is investigated in Section 5 while, in Section 6, a number of further related distributions are briefly considered. It should be the case that similarities and differences between the beta and Kumaraswamy’s distributions are made clear as the paper progresses but, in any case, they are summarised and discussed a little more in the closing Section 7.

For references to the hydrological literature on Kumaraswamy’s distribution, see [26]. The current article gives a much more complete account of the properties of the Kumaraswamy distribution than any previous publication, however.

Note that the linear transformation $\ell +(u-\ell )X$ moves a random variable $X$ on $(0,1)$ to any other bounded support $(\ell ,u)$. So, provided $\ell$ and $u$ don’t depend on $\alpha$ or $\beta$ and are known, there is no need to mention such an extension further.