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 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 [0,1] 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
f ( x ; a , b ) = a b x a − 1 ( 1 − x a ) b − 1 , where x ∈ ( 0 , 1 ) , {\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 The cumulative distribution function is
F ( x ; a , b ) = ∫ 0 x f ( ξ ; a , b ) d ξ = 1 − ( 1 − x a ) b . {\displaystyle 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 ) 1 b ) 1 a . {\displaystyle 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 = z − z min z max − z min , z min ≤ z ≤ z max . {\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:
m n = b Γ ( 1 + n / a ) Γ ( b ) Γ ( 1 + b + n / a ) = b B ( 1 + n / a , b ) {\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:
σ 2 = m 2 − m 1 2 . {\displaystyle \sigma ^{2}=m_{2}-m_{1}^{2}.}
The Shannon entropy (in nats) of the distribution is:
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