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 Ψ_{ρ,u}(y):=\pk{ρS_1+ \sqrt{1- ρ^2} S_2> y \lvert S_1> u}, ρ\in (-1,1),\in R when u approaches the upper endpoint of F. On the density function of Θwe 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 Ψ_{ρ,u}, which is then utilised to construct two estimators for the conditional distribution function 1- Ψ_{ρ,u}. Further, we allow Θto depend on u.

14 pages, paper submitted to Extremes in 2007

References in corpus (1)

• Estimation of bivariate excess probabilities for elliptical models

Cited by in corpus (7)

• Asymptotics of Random Contractions
• On the residual dependence index of elliptical distributions
• Exact Asymptotics of Bivariate Scale Mixture Distributions
• Minima and maxima of elliptical arrays and spherical processes
• Limit conditional distributions for bivariate vectors with polar representation
• Estimation of conditional laws given an extreme component
• Conditional Limits of W_p scale Mixture Distributions