On the strong Kotz approximation of Dirichlet random vectors

Abstract

Let (X 1, X 2) be a bivariate L p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α1,α2. If p=α1=α2=2, then (X 1, X 2) is a spherical random vector. The estimation of the conditional distribution of Z u *:=X 2 | X 1>u for u large is of some interest in statistical applications. When (X 1, X 2) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z u :=X 2| X 1=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z u * and Z u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.