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| {{Other uses|Kumaraswamy (disambiguation)}}
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| {{Probability distribution |
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| name =Kumaraswamy|
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| type =density|
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| pdf_image =[[Image:KumaraswamyT pdf.svg|325px|Probability density function]]|
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| cdf_image =[[Image:Kumaraswamy cdf.svg|325px|Cumulative distribution function]]|
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| parameters =<math>a>0\,</math> [[real number|(real)]]<br><math>b>0\,</math> [[real number|(real)]]|
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| support =<math>x \in [0,1]\,</math>|
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| pdf =<math>abx^{a-1}(1-x^a)^{b-1}\,</math>|
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| cdf =<math>[1-(1-x^a)^b]\,</math>|
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| mean =<math>\frac{b\Gamma(1+\tfrac{1}{a})\Gamma(b)}{\Gamma(1+\tfrac{1}{a}+b)}\,</math>|
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| median =<math>\left(1-2^{-1/b}\right)^{1/a}</math>|
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| mode =<math>\left(\frac{a-1}{ab-1}\right)^{1/a}</math> for <math>a\geq 1, b\geq 1, (a,b)\neq (1,1)</math>|
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| variance =(complicated-see text)|
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| skewness =(complicated-see text)|
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| kurtosis =(complicated-see text)|
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| entropy =|
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| mgf =|
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| char =
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| }}
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| In [[probability]] and [[statistics]], the '''Kumaraswamy's double bounded distribution''' is a family of [[continuous probability distribution]]s defined on the interval [0,1] differing in the values of their two non-negative [[shape parameter]]s, ''a'' and ''b''.
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| It is similar to the [[Beta distribution]], but much simpler to use especially in simulation studies due to the simple [[closed-form solution|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.
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| ==Characterization==
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| ===Probability density function===
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| The [[probability density function]] of the Kumaraswamy distribution is
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| :<math> f(x; a,b) = a b x^{a-1}{ (1-x^a)}^{b-1}.</math>
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| ===Cumulative distribution function===
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| The [[cumulative distribution function]] is therefore
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| :<math>F(x; a,b)=1-(1-x^a)^b.\ </math>
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| ===Generalizing to arbitrary range===
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| 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:
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| :<math> x = \frac{z-z_{\text{min}}}{z_{\text{max}}-z_{\text{min}}} , \qquad z_{\text{min}} \le z \le z_{\text{max}}. \,\!</math>
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| ==Properties==
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| The raw [[moment (mathematics)|moments]] of the Kumaraswamy distribution are given by {{Citation needed|date=June 2008}}:
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| :<math>m_n = \frac{b\Gamma(1+n/a)\Gamma(b)}{\Gamma(1+b+n/a)} = bB(1+n/a,b)\,</math>
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| where ''B'' is the [[Beta function]]. The variance, skewness, and excess kurtosis can be calculated from these raw moments. For example, the variance is:
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| :<math>\sigma^2=m_2-m_1^2.</math>
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| ==Relation to the Beta distribution==
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| The Kuramaswamy distribution is closely related to Beta distribution.
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| Assume that ''X''<sub>a,b</sub> is a Kumaraswamy distributed random variable with parameters ''a'' and ''b''.
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| Then ''X''<sub>a,b</sub> is the ''a''-th root of a suitably defined Beta distributed random variable.
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| More formally, Let ''Y''<sub>1,b</sub> denote a [[Beta distribution|Beta distributed]] random variable with parameters <math>\alpha=1</math> and <math>\beta=b</math>.
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| One has the following relation between ''X''<sub>a,b</sub> and ''Y''<sub>1,b</sub>.
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| :<math>X_{a,b}=Y^{1/a}_{1,b},</math>
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| with equality in distribution.
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| :<math>\operatorname{P}\{X_{a,b}\le x\}=\int_0^x ab t^{a-1}(1-t^a)^{b-1}dt=
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| \int_0^{x^a} b(1-t)^{b-1}dt=\operatorname{P}\{Y_{1,b}\le x^a\}
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| =\operatorname{P}\{Y^{1/a}_{1,b}\le x\}
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| .</math>
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| One may introduce generalised Kuramaswamy distributions by considering random variables of the form
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| <math>Y^{1/\gamma}_{\alpha,\beta}</math>, with <math>\gamma>0</math> and where <math>Y_{\alpha,\beta}</math>
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| denotes a Beta distributed random variable with parameters <math>\alpha</math> and <math>\beta</math>.
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| The raw [[moment (mathematics)|moments]] of this generalized Kumaraswamy distribution are given by:
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| :<math>m_n = \frac{\Gamma(\alpha+\beta)\Gamma(\alpha+n/\gamma)}{\Gamma(\alpha)\Gamma(\alpha+\beta+n/\gamma)}.</math> | |
| Note that we can reobtain the original moments setting <math>\alpha=1</math>, <math>\beta=b</math> and <math>\gamma=a</math>.
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| However, in general the cumulative distribution function does not have a closed form solution.
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| ==Related distributions==
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| *If <math> X \sim \textrm{Kumaraswamy}(1,1)\,</math> then <math> X \sim U(0,1)\,</math>
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| *If <math> X \sim U(0,1) \,</math> ([[Uniform distribution (continuous)]]) then <math> {\left( 1 - {\left( 1-X \right) }^{\tfrac{1}{b}} \right) }^{ \tfrac{1}{a} } \sim \textrm{Kumaraswamy}(a,b)\,</math>
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| *If <math> X \sim \textrm{Beta}(1,b) \, </math> ([[Beta distribution]]) then <math> X \sim \textrm{Kumaraswamy}(1,b)\, </math>
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| *If <math> X \sim \textrm{Beta}(a,1) \, </math> ([[Beta distribution]]) then <math> X \sim \textrm{Kumaraswamy}(a,1)\, </math>
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| *If <math> X \sim \textrm{Kumaraswamy}(a,1)\,</math> then <math> (1-X) \sim \textrm{Kumaraswamy}(1, a)\,</math>
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| *If <math> X \sim \textrm{Kumaraswamy}(1,a)\,</math> then <math> (1-X) \sim \textrm{Kumaraswamy}(a, 1)\,</math>
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| *If <math> X \sim \textrm{Kumaraswamy}(a,1)\,</math> then <math> -ln(X) \sim \textrm{Exponential}(a)\,</math>
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| *If <math> X \sim \textrm{Kumaraswamy}(1,b)\,</math> then <math> -ln(1-X) \sim \textrm{Exponential}(b)\,</math>
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| *If <math> X \sim \textrm{Kumaraswamy}(a,b)\,</math> then <math> X \sim \textrm{GB1}(a, 1, 1, b)\,</math>, the [[Generalized_Beta_distribution#Related distributions|generalized beta distribution of the first kind]].
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| ==Example==
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| A good example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity ''z''<sub>max</sub> whose upper bound is ''z''<sub>max</sub> and lower bound is 0 (Fletcher & Ponnambalam, 1996).
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| ==References==
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| * {{cite journal | doi=10.1016/0022-1694(80)90036-0 | author=Kumaraswamy, P. | title=A generalized probability density function for double-bounded random processes | journal=[[Journal of Hydrology]] | year=1980 | volume=46 | issue=1-2 | pages= 79–88}}
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| * {{cite journal | doi=10.1016/0022-1694(95)02946-X | author=Fletcher, S.G., and Ponnambalam, K. | title=Estimation of reservoir yield and storage distribution using moments analysis | journal=Journal of Hydrology | year=1996 | volume=182 | issue=1-4 | pages=259–275}}
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| * {{cite journal | doi=10.1016/j.stamet.2008.04.001 | author=Jones, M.C. | title=Kumaraswamy's distribution: A beta-type distribution with some tractability advantages | journal=Statistical Methodology | year=2009 | volume=6 | issue=1 | pages=70–81}}
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| * {{cite journal | doi=10.1080/00949655.2010.511621 | author=Lemonte, A.J. | title=Improved point estimation for the Kumaraswamy distribution | journal=Journal of Statistical Computation and Simulation | year=2011 | volume=81 | issue=12 | pages=1971–1982}}
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| {{ProbDistributions|continuous-bounded}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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