Conditional limit results for type I polar distributions

Abstract

Let (S ₁,S ₂) = (R cos(Θ), R sin(Θ)) be a bivariate random vector with associated random radius R which has distribution function F being further independent of the random angle Θ. In this paper we investigate the asymptotic behaviour of the conditional survivor probability $\overline{I}_{\rho,u}(y):=\mbox{\rm$\boldsymbol{P}$} \left\{{\rho S_1+ \sqrt{1- \rho^2} S_2> y \lvert S_1> u}\right\} , \rho \in (-1,1),\in \!I\!\!R$ when u approaches the upper endpoint of F. On the density function of Θ we impose 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 $\overline{I}_{\rho,u}$, which is then utilised to construct two estimators for the conditional distribution function $1- \overline{I}_{\rho,u}$. Furthermore, we allow Θ to depend on u.