Conditional Limit Results for Type I Polar Distributions

TL;DR

This paper studies the asymptotic behavior of conditional survivor probabilities for bivariate polar distributions, providing expansions and estimators under Gumbel max-domain conditions and local angular density assumptions.

Contribution

It introduces new asymptotic expansions for conditional survivor probabilities and develops estimators for the conditional distribution in polar distributions.

Findings

Derived asymptotic expansion of alpha_{ ho,u}

Constructed two estimators for the conditional distribution function

Allowed alpha to depend on u

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.