Estimation of conditional laws given an extreme component

TL;DR

This paper develops estimators for the limiting conditional distribution and quantile functions of a bivariate vector given an extreme component, with proven consistency and a CLT, supported by simulations.

Contribution

It introduces new estimators for the limiting distribution and quantile functions under extreme conditioning, with theoretical guarantees and simulation validation.

Findings

Estimators are consistent under the model assumptions.

A functional central limit theorem is established for the estimators.

Simulation studies demonstrate good small-sample performance.

Abstract

Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Y\leq y \mid X >t) $ is challenging when $t$ is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector $(X,Y)$, suitably normalized, given that $X$ is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as well as an estimator of the conditional quantile function when the conditioning event is extreme. Consistency of the estimators is proved and a functional central limit theorem for the estimator of the limiting distribution is obtained. The small sample behavior of the estimator of the conditional quantile function is illustrated through simulations.