# Kumaraswamy distribution

The Kumaraswamy distribution is a hidden gem in the world of probability distributions. This two-variable family of distributions is incredibly versatile, allowing you to model a variety of shapes with relative ease. The distribution is particularly suited to natural phenomena with outcomes bounded from both sides, including heights of people, hydrological daily data of rain fall temperatures, and test scores.

## Kumaraswamy distribution properties

The Kumaraswamy distribution was originally proposed by Indian hydrologist Poondi Kumaraswamy in 1980 [1] for variables with lower and upper bounds with a zero-inflation. It packs a punch with its straightforward probability density function (PDF) and cumulative distribution function (CDF) that can be easily expressed.

The Kumaraswamy distribution PDF can be expressed, in closed form and without consideration for inflation, as

where a and b are non-negative shape parameters.

The PDF can take on a number of different shapes including bathtub shaped, unimodal, uniantimodal, monotone increasing, and monotone decreasing depending on the values of its parameters. In some cases, singularities exist at the boundaries of the domain.

The CDF is invertible and has the closed form

P(x) = 1 – (1 – xa)b).

The quantile function is

F-1(p) = 1 – (1- p1/b)1/a

Kumaraswamy’s original work contained bounds at (0, 1). The bounds were extended to include inflations at both extremes [0,1] by S. G . Fletcher in 1996 [2].

## Comparison with beta distribution

The Kumaraswamy distribution is similar in shape to the beta distribution and is often called a “Beta-like” distribution. This should come as no surprise: Kumaraswamy derived his distribution from the beta distribution after fixing some parameters [3].

Unlike the fairly tractable beta distribution, the Kumaraswamy distribution CDF doesn’t contain the incomplete Beta function, which makes it easier to work with [4]. In addition, the closed form of the PDF and quantile functions make the Kumaraswamy distribution a more practical choice for many applications—especially simulation studies. Other advantages over the beta include explicit formulae for moments of order statistics and a simple formula for generation of random variables [5].

In sum, the Kumaraswamy distribution shines in its ease of use and tractability [6] in certain situations. It’s a valuable tool that shouldn’t be overlooked in your data analysis toolbox.

## References

[1] Kumaraswamy, P. (1980). “A generalized probability density function for double-bounded random processes”. Journal of Hydrology46 (1–2): 79–88. Bibcode:1980JHyd…46…79Kdoi:10.1016/0022-1694(80)90036-0ISSN 0022-1694.

[2] Fletcher, S.G.; Ponnambalam, K. (1996). “Estimation of reservoir yield and storage distribution using moments analysis”. Journal of Hydrology182 (1–4): 259–275. Bibcode:1996JHyd..182..259Fdoi:10.1016/0022-1694(95)02946-xISSN 0022-1694.

[3]

[4] Michalowicz, J. et al., (2013). Handbook of Differential Entropy. CRC Press.

[5] Ishaq et al., (2019). On some properties of Generalized Transmuted Kumaraswamy distribution. Pakistan Journal of Statistics and Operation Research; Lahore Vol. 15, Iss. 3, (2019): 577-586.

[6] M.C. Jones, Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages, Statistical Methodology, Volume 6, Issue 1, 2009,
Pages 70-81, ISSN 1572-3127, https://doi.org/10.1016/j.stamet.2008.04.001.

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