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On the strong Kotz approximation of Dirichlet random vectors
Let (X1, X2) be a bivariate Lp-norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters 1,2. If p=1=2=2, then (X1, X2) is a spherical random vector. The estimation of the conditional distribution of Zu*:=X2 | X1u for u large is of some interest in statistical applications. When (X1, X2) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Zu* can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Zu:=X2| X1=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 Zu* and Zu 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.
Lp-norm generalized symmetrized Dirichlet distribution, Conditional limit theorem, Kotz type I LpGSD distribution, Gumbel max-domain of attraction, Concomitants of order statistics, Estimation of conditional quantile function
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