Kumaraswamy distribution

For other uses, see Kumaraswamy (disambiguation).

Kumaraswamy
Probability density function
Cumulative distribution function
Parameters	$a>0\,$ (real) $b>0\,$ (real)
Support	$x\in [0,1]\,$
PDF	$abx^{{a-1}}(1-x^{a})^{{b-1}}\,$
CDF	$[1-(1-x^{a})^{b}]\,$
Mean	${\frac {b\Gamma (1+{\tfrac {1}{a}})\Gamma (b)}{\Gamma (1+{\tfrac {1}{a}}+b)}}\,$
Median	$\left(1-2^{{-1/b}}\right)^{{1/a}}$
Mode	$\left({\frac {a-1}{ab-1}}\right)^{{1/a}}$ for $a\geq 1,b\geq 1,(a,b)\neq (1,1)$
Variance	(complicated-see text)
Skewness	(complicated-see text)
Ex. kurtosis	(complicated-see text)
Entropy	$\left(1\!-\!{\tfrac {1}{a}}\right)+\left(1\!-\!{\tfrac {1}{b}}\right)H_{b}+\ln(ab)$

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 due to the simple closed form of both its probability density function and cumulative distribution function. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded.

Characterization

Probability density function

The probability density function of the Kumaraswamy distribution is

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)=\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:

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

where B is the Beta function. The variance, skewness, and excess kurtosis can be calculated from these raw moments. For example, the variance is:

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

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

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

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

Relation to the Beta distribution

The Kumaraswamy distribution is closely related to Beta distribution. 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 $\alpha =1$ and $\beta =b$ . One has the following relation between X_a,b and Y_1,b.

X_{{a,b}}=Y_{{1,b}}^{{1/a}},

with equality in distribution.

\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 $Y_{{\alpha ,\beta }}^{{1/\gamma }}$ , with $\gamma >0$ and where $Y_{{\alpha ,\beta }}$ denotes a Beta distributed random variable with parameters $\alpha$ and $\beta$ . The raw moments of this generalized Kumaraswamy distribution are given by:

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

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

Related distributions

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

Example

A good example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity z_max whose upper bound is z_max and lower bound is 0 (Fletcher & Ponnambalam, 1996).

References

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.
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.
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.
Lemonte, A.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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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