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 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.^[2] A good example for inflations at extremes are the probabilities of full and empty reservoirs and are important for reservoir design.

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

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

The cumulative distribution function is

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

The inverse cumulative distribution function (quantile function) is

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

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:

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

The raw moments of the Kumaraswamy distribution are given by:^{[3] [4]}

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

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:

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

The Shannon entropy (in nats) of the distribution is:^[5]

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

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

The Kumaraswamy distribution is closely related to Beta distribution.^[6] Assume that X_a,b is a Kumaraswamy distributed random variable with parameters a and b. Then X_a,b is the a-th root of a suitably defined Beta distributed random variable. More formally, Let Y_1,b denote a Beta distributed random variable with parameters ${\displaystyle \alpha =1}${\displaystyle \alpha =1} and ${\displaystyle \beta =b}${\displaystyle \beta =b}. One has the following relation between X_a,b and Y_1,b.

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

with equality in distribution.

${\displaystyle \operatorname {P} \{X_{a,b}\leq x\}=\int _{0}^{x}abt^{a-1}(1-t^{a})^{b-1}dt=\int _{0}^{x^{a}}b(1-t)^{b-1}dt=\operatorname {P} \{Y_{1,b}\leq x^{a}\}=\operatorname {P} \{Y_{1,b}^{1/a}\leq x\}.}${\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 ${\displaystyle Y_{\alpha ,\beta }^{1/\gamma }}${\displaystyle Y_{\alpha ,\beta }^{1/\gamma }}, with ${\displaystyle \gamma >0}${\displaystyle \gamma >0} and where ${\displaystyle Y_{\alpha ,\beta }}${\displaystyle Y_{\alpha ,\beta }} denotes a Beta distributed random variable with parameters ${\displaystyle \alpha }${\displaystyle \alpha } and ${\displaystyle \beta }${\displaystyle \beta }. The raw moments of this generalized Kumaraswamy distribution are given by:

${\displaystyle m_{n}={\frac {\Gamma (\alpha +\beta )\Gamma (\alpha +n/\gamma )}{\Gamma (\alpha )\Gamma (\alpha +\beta +n/\gamma )}}.}${\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 \alpha =1}, ${\displaystyle \beta =b}${\displaystyle \beta =b} and ${\displaystyle \gamma =a}${\displaystyle \gamma =a}. However, in general, the cumulative distribution function does not have a closed form solution.

• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,1),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,1),円} then ${\displaystyle X\sim U(0,1),円}${\displaystyle X\sim U(0,1),円} (Uniform distribution)
• If ${\displaystyle X\sim U(0,1),円}${\displaystyle X\sim U(0,1),円} then ${\displaystyle {\left(1-X^{\tfrac {1}{b}}\right)}^{\tfrac {1}{a}}\sim {\textrm {Kumaraswamy}}(a,b),円}${\displaystyle {\left(1-X^{\tfrac {1}{b}}\right)}^{\tfrac {1}{a}}\sim {\textrm {Kumaraswamy}}(a,b),円}^[6]
• If ${\displaystyle X\sim {\textrm {Beta}}(1,b),円}${\displaystyle X\sim {\textrm {Beta}}(1,b),円} (Beta distribution) then ${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,b),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,b),円}
• If ${\displaystyle X\sim {\textrm {Beta}}(a,1),円}${\displaystyle X\sim {\textrm {Beta}}(a,1),円} (Beta distribution) then ${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1),円}
• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1),円} then ${\displaystyle (1-X)\sim {\textrm {Kumaraswamy}}(1,a),円}${\displaystyle (1-X)\sim {\textrm {Kumaraswamy}}(1,a),円}
• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,a),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,a),円} then ${\displaystyle (1-X)\sim {\textrm {Kumaraswamy}}(a,1),円}${\displaystyle (1-X)\sim {\textrm {Kumaraswamy}}(a,1),円}
• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1),円} then ${\displaystyle -\log(X)\sim {\textrm {Exponential}}(a),円}${\displaystyle -\log(X)\sim {\textrm {Exponential}}(a),円}
• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,b),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(1,b),円} then ${\displaystyle -\log(1-X)\sim {\textrm {Exponential}}(b),円}${\displaystyle -\log(1-X)\sim {\textrm {Exponential}}(b),円}
• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,b),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,b),円} then ${\displaystyle X\sim {\textrm {GB1}}(a,1,1,b),円}${\displaystyle X\sim {\textrm {GB1}}(a,1,1,b),円}, the generalized beta distribution of the first kind
• If ${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,b),円}${\displaystyle X\sim {\textrm {Kumaraswamy}}(a,b),円} then ${\displaystyle \left\{{\frac {1}{1-X}}\right\}\sim {\textrm {MK}}(a,b),円}${\displaystyle \left\{{\frac {1}{1-X}}\right\}\sim {\textrm {MK}}(a,b),円}, the Modified Kumaraswamy distribution.

An example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity z whose upper bound is z_max 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.^[2]

• Continuous distributions

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