The inverse cumulative distribution function (quantile function) is
Generalizing to arbitrary interval supportedit
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:
Propertiesedit
The raw moments of the Kumaraswamy distribution are given by:[3][4]
The Kumaraswamy distribution is closely related to Beta distribution.[6]
Assume that Xa,b is a Kumaraswamy distributed random variable with parameters a and b.
Then Xa,b is the a-th root of a suitably defined Beta distributed random variable.
More formally, Let Y1,b denote a Beta distributed random variable with parameters and .
One has the following relation between Xa,b and Y1,b.
with equality in distribution.
One may introduce generalised Kumaraswamy distributions by considering random variables of the form
, with and where
denotes a Beta distributed random variable with parameters and .
The raw moments of this generalized Kumaraswamy distribution are given by:
Note that we can re-obtain the original moments setting , and .
However, in general, the cumulative distribution function does not have a closed form solution.
An example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity z whose upper bound is zmax 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]
Referencesedit
^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.
^ abFletcher, 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" (PDF). Anais da Academia Brasileira de Ciências. 91 (2): e20180955. doi:10.1590/0001-3765201920180955. ISSN 1678-2690. PMID 31141016. S2CID 169034252.
^Michalowicz, Joseph Victor; Nichols, Jonathan M.; Bucholtz, Frank (2013). Handbook of Differential Entropy. Chapman and Hall/CRC. p. 100. ISBN 9781466583177.
^ abJones, 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.