Kumaraswamy distribution

Kumaraswamy distribution

Probability distribution
name =Kumaraswamy
type =density
pdf_

cdf_

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{bGamma(1+1/a)Gamma(b)}{Gamma(1+1/a+b)},
median =left(1-left(frac{1}{2} ight)^{1/b} ight)^{1/a}
mode =left(frac{a-1}{ab-1} ight)^{1/a}
variance =(complicated-see text)
skewness =(complicated-see text)
kurtosis =(complicated-see text)
entropy =
mgf =
char =
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) = a b x^{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, we may replace the normalized variable "x" with the unshifted and unscaled variable "z" where:

: x = frac{z-z_{mathrm{min}{z_{mathrm{max-z_{mathrm{min} , qquad z_{mathrm{min le z le z_{mathrm{max. ,!

The distribution is sometimes combined with a "pike probability" or a Dirac delta function, e.g.:

: g(x|a,b) = F_0delta(x)+(1-F_0)a b x^{a-1}{ (1-x^a)}^{b-1}.

Properties

The raw moments of the Kumaraswamy distribution are given by Fact|date=June 2008:

:m_n = frac{bGamma(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 eta=b. One has the following relation between "X"a,b and "Y"1,b.

:X_{a,b}=Y^{1/a}_{1,b},

with equality in distribution.

:operatorname{P}{X_{a,b}le x}=int_0^x ab t^{a-1}(1-t^a)^{b-1}dt=int_0^{x^a} b(1-t)^{b-1}dt=operatorname{P}{Y_{1,b}le x^a}=operatorname{P}{Y^{1/a}_{1,b}le x}.

One may introduce generalised Kuramaswamy distributions by considering random variables of the formY^{1/gamma}_{alpha,eta}, with gamma>0 and where Y_{alpha,eta} denotes a Beta distributed random variable with parameters alpha and eta.The raw moments of this generalized Kumaraswamy distribution are given by::m_n = frac{Gamma(alpha+eta)Gamma(alpha+n/gamma)}{Gamma(alpha)Gamma(alpha+eta+n/gamma)}.Note that we can reobtain the original moments setting alpha=1, eta=b and gamma=a.However, in general the cumulative distribution function does not have a closed form solution.

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, 1996).

References

*
*


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Kumaraswamy — or Kumaraswami may refer to:*H. D. Kumaraswamy, former Chief Minister of the state of Karnataka, India *Poondi Kumaraswamy (1930–1988), Indian engineer and hydrologist *Murugan, also called Kumaraswami , most popular Hindu deity amongst Tamils of …   Wikipedia

  • Poondi Kumaraswamy — Ponnambalam Kumaraswamy (often referred to as Poondi Kumaraswamy) was a leading hydrologist from India[1][2]. He was elected a Fellow of the Indian Academy of Sciences in 1972[3] although his only formal education was a Civil Engineering degree… …   Wikipedia

  • Beta distribution — Probability distribution name =Beta| type =density pdf cdf parameters =alpha > 0 shape (real) eta > 0 shape (real) support =x in [0; 1] ! pdf =frac{x^{alpha 1}(1 x)^{eta 1 {mathrm{B}(alpha,eta)}! cdf =I x(alpha,eta)! mean… …   Wikipedia

  • Cauchy distribution — Not to be confused with Lorenz curve. Cauchy–Lorentz Probability density function The purple curve is the standard Cauchy distribution Cumulative distribution function …   Wikipedia

  • Maxwell–Boltzmann distribution — Maxwell–Boltzmann Probability density function Cumulative distribution function parameters …   Wikipedia

  • Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function …   Wikipedia

  • Probability distribution — This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics). For other uses, see Distribution (disambiguation). In probability theory, a probability mass, probability density …   Wikipedia

  • Negative binomial distribution — Probability mass function The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation. notation: parameters: r > 0 number of failures until the experiment is stopped (integer,… …   Wikipedia

  • Exponential distribution — Not to be confused with the exponential families of probability distributions. Exponential Probability density function Cumulative distribution function para …   Wikipedia

  • Chi-squared distribution — This article is about the mathematics of the chi squared distribution. For its uses in statistics, see chi squared test. For the music group, see Chi2 (band). Probability density function Cumulative distribution function …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”