Scale mixtures of Kotz–Dirichlet distributions

Abstract

a b s t r a c tIn this paper, we first show that a k-dimensional Dirichlet random vector has independent components if and only if it is a Kotz Type I Dirichlet random vector. We then consider in detail the class of k-dimensional scale mixtures of Kotz-Dirichlet random vectors, which is a natural extension of the class of Kotz Type I random vectors. An interesting member of the Kotz-Dirichlet class of multivariate distributions is the family of Pearson-Kotz Dirichlet distributions, for which we present a new distributional property. In an asymptotic framework, we show that the Kotz Type I Dirichlet distributions approximate the conditional distributions of scale mixtures of Kotz-Dirichlet random vectors. Furthermore, we show that the tail indices of regularly varying Dirichlet random vectors can be expressed in terms of the Kotz Type I Dirichlet random vectors.

Extremes of aggregated Dirichlet risks

“…, U d ) are mutually independent. Basic distributional and asymptotic properties of Dirichlet random vectors are discussed in numerous contributions; see e.g., [11,21,4,22,23,27,1,2] and the references therein.…”

“…for some positive scaling function w (here x F is the upper endpoint of F and we abbreviate (2) as F ∈ GMDA(w)). We use in (2) the standard notation ∼ for the asymptotic equivalence of two real-valued functions.…”

“…for some positive scaling function w (here x F is the upper endpoint of F and we abbreviate (2) as F ∈ GM DA(w)).…”

“…g., [2]. Such a random vector has stochastic representation (1) with R having Gamma(α, 1) distribution.…”

Tail Behavior of Weighted Sums of Order Statistics of Dependent Risks

“…where " Davis and Resnick (1988) or relation (5) of Balakrishnan and Hashorva (2013). Namely, for any distribution function F ∈ GMDA(h) with an infinite upper endpoint, the following bound holds for all large x and ε > 0:…”