Kumaraswamy
Probability density function
Cumulative distribution function
Parameters
a
>
0
{\displaystyle a>0\,}
(real)
b
>
0
{\displaystyle b>0\,}
(real) Support
x
∈
(
0
,
1
)
{\displaystyle x\in (0,1)\,}
PDF
a
b
x
a
−
1
(
1
−
x
a
)
b
−
1
{\displaystyle abx^{a-1}(1-x^{a})^{b-1}\,}
CDF
1
−
(
1
−
x
a
)
b
{\displaystyle 1-(1-x^{a})^{b}}
Mean
b
Γ
(
1
+
1
a
)
Γ
(
b
)
Γ
(
1
+
1
a
+
b
)
{\displaystyle {\frac {b\Gamma (1+{\tfrac {1}{a}})\Gamma (b)}{\Gamma (1+{\tfrac {1}{a}}+b)}}\,}
Median
(
1
−
2
−
1
/
b
)
1
/
a
{\displaystyle \left(1-2^{-1/b}\right)^{1/a}}
Mode
(
a
−
1
a
b
−
1
)
1
/
a
{\displaystyle \left({\frac {a-1}{ab-1}}\right)^{1/a}}
a
≥
1
,
b
≥
1
,
(
a
,
b
)
≠
(
1
,
1
)
{\displaystyle 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
(
1
−
1
b
)
+
(
1
−
1
a
)
H
b
−
ln
(
a
b
)
{\displaystyle \left(1\!-\!{\tfrac {1}{b}}\right)+\left(1\!-\!{\tfrac {1}{a}}\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 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] [2]
Characterization [ edit ] Probability density function [ edit ] 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 [ edit ] 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 [ edit ] The inverse cumulative distribution function (quantile function) is
F
(
y
;
a
,
b
)
−
1
=
(
1
−
(
1
−
y
)
1
b
)
1
a
.
{\displaystyle F(y;a,b)^{-1}=(1-(1-y)^{\frac {1}{b}})^{\frac {1}{a}}.\ }
Generalizing to arbitrary interval support [ edit ] 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 [ edit ] The raw moments of the Kumaraswamy distribution are given by[3] [4]
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:[5]
H
=
(
1
−
1
b
)
+
(
1
−
1
a
)
H
b
−
ln
(
a
b
)
{\displaystyle H=\left(1\!-\!{\tfrac {1}{b}}\right)+\left(1\!-\!{\tfrac {1}{a}}\right)H_{b}-\ln(ab)}
where
H
i
{\displaystyle H_{i}}
harmonic number function.
Relation to the Beta distribution [ edit ] The Kumaraswamy distribution is closely related to Beta distribution.[6] 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
α
=
1
{\displaystyle \alpha =1}
β
=
b
{\displaystyle \beta =b}
X a,b and Y 1,b .
X
a
,
b
=
Y
1
,
b
1
/
a
,
{\displaystyle X_{a,b}=Y_{1,b}^{1/a},}
with equality in distribution.
P
{
X
a
,
b
≤
x
}
=
∫
0
x
a
b
t
a
−
1
(
1
−
t
a
)
b
−
1
d
t
=
∫
0
x
a
b
(
1
−
t
)
b
−
1
d
t
=
P
{
Y
1
,
b
≤
x
a
}
=
P
{
Y
1
,
b
1
/
a
≤
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
Y
α
,
β
1
/
γ
{\displaystyle Y_{\alpha ,\beta }^{1/\gamma }}
γ
>
0
{\displaystyle \gamma >0}
Y
α
,
β
{\displaystyle Y_{\alpha ,\beta }}
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
moments of this generalized Kumaraswamy distribution are given by:
m
n
=
Γ
(
α
+
β
)
Γ
(
α
+
n
/
γ
)
Γ
(
α
)
Γ
(
α
+
β
+
n
/
γ
)
.
{\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
α
=
1
{\displaystyle \alpha =1}
β
=
b
{\displaystyle \beta =b}
γ
=
a
{\displaystyle \gamma =a}
Related distributions [ edit ] If
X
∼
Kumaraswamy
(
1
,
1
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(1,1)\,}
X
∼
U
(
0
,
1
)
{\displaystyle X\sim U(0,1)\,}
If
X
∼
U
(
0
,
1
)
{\displaystyle X\sim U(0,1)\,}
Uniform distribution (continuous) ) then
(
1
−
(
1
−
X
)
1
b
)
1
a
∼
Kumaraswamy
(
a
,
b
)
{\displaystyle {{\Big (}1-{\left(1-X\right)}^{\tfrac {1}{b}}{\Big )}}^{\tfrac {1}{a}}\sim {\textrm {Kumaraswamy}}(a,b)\,}
If
X
∼
Beta
(
1
,
b
)
{\displaystyle X\sim {\textrm {Beta}}(1,b)\,}
Beta distribution ) then
X
∼
Kumaraswamy
(
1
,
b
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(1,b)\,}
If
X
∼
Beta
(
a
,
1
)
{\displaystyle X\sim {\textrm {Beta}}(a,1)\,}
Beta distribution ) then
X
∼
Kumaraswamy
(
a
,
1
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1)\,}
If
X
∼
Kumaraswamy
(
a
,
1
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1)\,}
(
1
−
X
)
∼
Kumaraswamy
(
1
,
a
)
{\displaystyle (1-X)\sim {\textrm {Kumaraswamy}}(1,a)\,}
If
X
∼
Kumaraswamy
(
1
,
a
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(1,a)\,}
(
1
−
X
)
∼
Kumaraswamy
(
a
,
1
)
{\displaystyle (1-X)\sim {\textrm {Kumaraswamy}}(a,1)\,}
If
X
∼
Kumaraswamy
(
a
,
1
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(a,1)\,}
−
log
(
X
)
∼
Exponential
(
a
)
{\displaystyle -\log(X)\sim {\textrm {Exponential}}(a)\,}
If
X
∼
Kumaraswamy
(
1
,
b
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(1,b)\,}
−
log
(
1
−
X
)
∼
Exponential
(
b
)
{\displaystyle -\log(1-X)\sim {\textrm {Exponential}}(b)\,}
If
X
∼
Kumaraswamy
(
a
,
b
)
{\displaystyle X\sim {\textrm {Kumaraswamy}}(a,b)\,}
X
∼
GB1
(
a
,
1
,
1
,
b
)
{\displaystyle X\sim {\textrm {GB1}}(a,1,1,b)\,}
generalized beta distribution of the first kind . Example [ edit ] 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]
References [ edit ]
^ Kumaraswamy, P. (1980). "A generalized probability density function for double-bounded random processes". Journal of Hydrology . 46 (1–2): 79–88. Bibcode :1980JHyd...46...79K . doi :10.1016/0022-1694(80)90036-0 . ISSN 0022-1694 . ^ a b Fletcher, S.G.; Ponnambalam, K. (1996). "Estimation of reservoir yield and storage distribution using moments analysis". Journal of Hydrology . 182 (1–4): 259–275. Bibcode :1996JHyd..182..259F . doi :10.1016/0022-1694(95)02946-x . ISSN 0022-1694 . ^ Lemonte, Artur 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 . ISSN 0094-9655 . ^ CRIBARI-NETO, FRANCISCO; SANTOS, JÉSSICA (2019). "Inflated Kumaraswamy distributions" . Anais da Academia Brasileira de Ciências . 91 (2): e20180955. doi :10.1590/0001-3765201920180955 ISSN 1678-2690 . PMID 31141016 . ^ Michalowicz, Joseph Victor; Nichols, Jonathan M.; Bucholtz, Frank (2013). Handbook of Differential Entropy . Chapman and Hall/CRC. p. 100. ISBN 9781466583177 ^ 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 . ISSN 1572-3127 .
Discrete univariate Discrete univariate Continuous univariate Continuous univariate Continuous univariate Continuous univariate Mixed continuous-discrete univariate Multivariate (joint) Directional Degenerate and singular Families
