Estimation of conditional laws given an extreme component

Abstract

Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Y\le y\mid X>t)$ 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.

32 pages, 5 figure

References in corpus (6)

• Conditioning on an extreme component: Model consistency with regular variation on cones
• Estimation of bivariate excess probabilities for elliptical models
• Detecting a conditional extrme value model
• Limit conditional distributions for bivariate vectors with polar representation
• Conditional Limit Results for Type I Polar Distributions
• Conditional Limits of W_p scale Mixture Distributions

Cited by in corpus (4)

• Conditioning on an extreme component: Model consistency with regular variation on cones
• Limit conditional distributions for bivariate vectors with polar representation
• Gaussian Approximation of Perturbed Chi-Square Risks
• Multivariate Tail Estimation: Conditioning on an extreme event