Conditioning exceedances on covariate processes

Abstract

This article concerns the statistical inference for the upper tail of the conditional distribution of a response variable Y given a covariate X = x based on n random vectors within the parametric extreme value framework. Pioneering work in this field was done by Smith (Stat Sci 4:367–393, 1989) and Smith and Shively (Atmos Environ 29:3489–3499, 1995). We propose to base the inference on a conditional distribution of the point process of exceedances given the point process of covariates. It is of importance that the conditional distribution merely depends on the conditional distribution of the response variable given the covariates. In the special case of Poisson processes such a result may be found in Reiss (1993). Our results are valid within the broader model where the response variables are conditionally independent given the covariates. It is numerically exemplified that the maximum likelihood principle leads to more accurate estimators within the conditional approach than in the previous one.