Kumaraswamy distribution

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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)

In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous probability distributions defined on the interval [0,1] differing in the values of their two non-negative shape parameters, a and b.

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

$f(x;a,b)=abx^{{a-1}}{(1-x^{a})}^{{b-1}}.$

Cumulative distribution function

The cumulative distribution function is therefore

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

Generalizing to arbitrary range

In its simplest form, the distribution has a range 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 ^{[citation needed]}:

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

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

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

Relation to the Beta distribution

The Kuramaswamy 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 Kuramaswamy 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., and 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

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
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
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete 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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