Local polynomial maximum likelihood estimation for Pareto-type distributions

Abstract

We discuss the estimation of the tail index of a heavy-tailed distribution when covariate information is available. The approach followed here is based on the technique of local polynomial maximum likelihood estimation. The generalized Pareto distribution is fitted locally to exceedances over a high specified threshold. The method provides nonparametric estimates of the parameter functions and their derivatives up to the degree of the chosen polynomial. Consistency and asymptotic normality of the proposed estimators will be proven under suitable regularity conditions. This approach is motivated by the fact that in some applications the threshold should be allowed to change with the covariates due to significant effects on scale and location of the conditional distributions. Using the asymptotic results we are able to derive an expression for the asymptotic mean squared error, which can be used to guide the selection of the bandwidth and the threshold. The applicability of the method will be demonstrated with a few practical examples.