A nonparametric estimator for the conditional tail index of Pareto-type distributions

Abstract

The tail index is an important parameter in the whole of extreme value theory. In this article, we consider the estimation of the tail index in the presence of a random covariate, where the conditional distribution of the variable of interest is of Pareto-type. More precisely, we use a logarithmic function to link the tail index to the nonlinear predictor induced by covariates, which forms the nonparametric tail index regression models. To estimate the unknown function, we develop an estimation procedure via a local likelihood method. Consistency and asymptotic normality of the estimated functions are established. Subsequently, these theoretical results are illustrated through simulated and real datasets.