Estimating the probability of a rare event

Abstract

Let $(X_{1},Y_{1}), (X_{2},Y_{2}),\dots,(X_{n},Y_{n})$ be a random sample from a bivariate distribution function $F$ which is in the domain of attraction of a bivariate extreme value distribution function $G$. A subset $C$ of $\mathbb{R}^{2}$ is given, which contains none of the observations. We shall give an asymptotic confidence interval for $\Pr((X_{i},Y_{i}) \in C)$ under certain conditions.