Kumaraswamy Distribution and Random Extrema

where Y~G and T~Kα,β. This generalization of F was proposed in Cordeiro and de Castro [2], where the authors developed its basic properties and presented generalizations of normal, Weibull, gamma, Gumbel, and inverse Gaussian distributions. Many other generalized distributions following this scheme have been developed since then, including recent works of de Pascoa et al. [3], Nadarajah et al. [4], Mameli [5] and Aryal and Zhang [6]. However, papers on


INTRODUCTION
There is a growing literature on generalized distributions based on Kumaraswamy distribution [1].They are obtained from a "base" distribution with the cumulative distribution function (CDF) F as the CDF G (a generalized version of F) viz.

(1)
where is the CDF of the Kumaraswamy distribution with parameters α, β > 0. The latter from now is denoted by K α,β .In terms of random variables, the relation (1) can be stated as where Y~G and T~K α,β .This generalization of F was proposed in Cordeiro and de Castro [2], where the authors developed its basic properties and presented generalizations of normal, Weibull, gamma, Gumbel, and inverse Gaussian distributions.Many other generalized distributions following this scheme have been developed since then, including recent works of de Pascoa et al. [3], Nadarajah et al. [4], Mameli [5] and Aryal and Zhang [6].However, papers on the topic focus mainly on rather elementary properties and do not provide any significant theoretical interpretation of the construction.One exception is an interpretation for integer-valued α and β through maxima and minima of independent and identically distributed (IID) random components (e.g., Jones [7], Nadarajah et al. [4], Nadarajah and Eljabri [8]).Indeed, assuming that α = m, β = n are positive integers, we find that x m is the CDF of the maximum of m IID standard uniform variables, with the corresponding survivor function (SF) being 1 -x m .Thus, the quantity (1 -x m ) n in ( 2) is the SF of the minimum of n such random variables, with H being the corresponding CDF.In other words, we have the following stochastic representation of T~K m,n, ( where {U i,j } are IID standard uniform variables.This property, discussed in Jones [7], motivated the name minimax for this distribution.Below we extend this interpretation to the general Kumaraswamy distribution as well as to its generalization (1).

MAIN RESULTS
To obtain a representation of a Kumaraswamy random variable in terms of min/max, we shall use the following basic result (e.g., Kozubowski and Podgórski, [9]), which relates the distributions of min/max of IID components with random number of terms to the relevant Probability Generating Function (PGF).Lemma 2.1.If N is an random variable, independent of the sequence {X i } of IID random variables with the CDF F, then the CDFs of the random variables A notable application of this result, discussed in Kozubowski and Podgórski [9], concerns a special case where the variable N = N α in (5) has the Sibuya distribution (Sibuya, [10]), given by the PGF (6) and the Probability Mass Function (PMF) where necessarily , with the boundary case α = 1 corresponding to a unit mass at n = 1.This variable represents the number of trials till the first success in an infinite sequence of independent Bernoulli trials, where the probability of success varies with the trial, and for the nth trial equals α/n.Here, because of the special form of the PGF of the Sibuya random variable N α , if the latter represents the random number of terms in Lemma 2.1, the CDFs of the random variables X and Y in (5) are given by α , respectively.This leads to our first result for the Kumaraswamy distribution with the parameters restricted to the unit interval (0, 1).The above result can be re-formulated for any Kumaraswamy generalized random variable obtained viz.(3), providing a meaningful interpretation of this construction in terms of maxima and minima of IID components with the "parent" CDF F. Proposition 2.2.Let N α (i) , be IID Sibuya variables with parameter , and independent of another Sibuya variable N β , with parameter .Further, for a given CDF F, let Y be a random variable with the CDF G defined by (1), where H = K α, β ; with .Then we have where the {X i,j } are IID random variables with CDF F, independent of N α (i) and N β .
Remark 2.1.Let B α, β denote beta distribution given by the PDF Then, if either α = 1 or β = 1, the Kumaraswamy distribution K α, β coincides with B α, β .Thus, Proposition 2.1 specialized to this case, leads to stochastic representations of such beta-distributed random variables in terms of maxima and minima, i.e. if for the {U i } are IID standard uniform random variables, independent of Sibuya-distributed N α and N β with we define then B 1 ~Bα, 1 and B 2 ~B1, β .Further, if either α = 1 or β = 1, the generalized distributions obtained viz.(1) are special cases of beta-generalized distributions popularized by Eugene et al. [11], arising when H in (1) is the CDF corresponding to (10).Thus, Proposition 2.2 specialized to this case shows that if a random variables Y is defined viz.
(3), where either T~B α, 1 or T~B 1, β with , then we have respectively, where the {X i } are IID random variables with CDF F, independent of Sibuya-distributed N α and N β .
An extension of Proposition 2.1 to the case where the parameters are no longer restricted to the unit interval is straightforward.To formulate the result, we shall split a positive number r into r = {r} + < r >, where We shall also need the following result taken from Kozubowski and Podgórski [9], where we use the standard convention that the min and max over an empty set are understood as ∞ and -∞, respectively.

Proposition 2.3. If X and Y have CDFs given by 1 -(1 -F(x))
r and [F(x)] r , respectively, where F is a CDF and , then they admit the stochastic representations where N < r > has the Sibuya distribution (7) with parameter α = < r > and is independent of the IID {X j } with the CDF F.
When we set r = β and apply the first representation in ( 14) to a Kumaraswamy random variable T~K α, β , , we obtain where N < β > has Sibuya distribution with parameter < β > and {X j } is IID with the CDF x α .In turn, when we set r = α and apply the second representation in (14) to each X j in (15), we obtain where the {U i,j } are IID standard uniform variables and the N < α > (j) are IID and Sibuya distributed with parameter < α >, independent of the {U i,j }.By combining ( 15) with ( 16), we obtain the following result.
Proposition 2.4.Let T~K α, β with general α, β > 0. Then we have where the {U i,j } are IID standard uniform variables, N < α > (j) and N < β > have Sibuya distributions with respective parameters < α > and < β >, and all the variables on the right-hand-side of ( 17) are mutually independent.Moreover, if both parameters of T~K α, β are positive integers, so that in (18) we have and in (19) we have , then both, ( 18) and ( 19), turn into the representation (4) derived by Jones [7].