Conditional Limit Results for Type I Polar Distributions

Abstract

Let (S_1,S_2)=(R \cos(\Theta), R \sin (\Theta)) be a bivariate random vector with associated random radius R which has distribution function $F$ being further independent of the random angle \Theta. In this paper we investigate the asymptotic behaviour of the conditional survivor probability \Psi_{\rho,u}(y):=\pk{\rho S_1+ \sqrt{1- \rho^2} S_2> y \lvert S_1> u}, \rho \in (-1,1),\in R when u approaches the upper endpoint of F. On the density function of \Theta we require a certain local asymptotic behaviour at 0, whereas for F we require that it belongs to the Gumbel max-domain of attraction. The main result of this contribution is an asymptotic expansion of \Psi_{\rho,u}, which is then utilised to construct two estimators for the conditional distribution function 1- \Psi_{\rho,u}. Further, we allow \Theta to depend on u.