On tail inference in iid settings with nonnegative extreme value index

TL;DR

This paper compares various tail estimators in iid settings with nonnegative extreme value index, focusing on tail probability and mean excess function estimation, using theoretical analysis and simulations.

Contribution

It provides a theoretical and numerical comparison of tail estimators based on extreme value theory and nonparametric smoothing for nonnegative extreme value index distributions.

Findings

Kernel estimators show competitive mean squared errors.

Generalized Pareto fitting performs well in finite samples.

Plug-in Hall estimators have consistent convergence rates.

Abstract

In extreme value inference it is a fundamental problem how the target value is required to be extreme by the extreme value theory. In iid settings this study both theoretically and numerically compares tail estimators, which are based on either or both of the extreme value theory and the nonparametric smoothing. This study considers tail probability estimation and mean excess function estimation. This study assumes that the extreme value index of the underlying distribution is nonnegative. Specifically, the Hall class or the Weibull class of distributions is supposed in order to obtain the convergence rates of the estimators. This study investigates the nonparametric kernel type estimators, the fitting estimators to the generalized Pareto distribution and the plug-in estimators of the Hall distribution, which was proposed by Hall and Weissman (1997). In simulation studies the mean squared errors of the estimators in some finite sample cases are compared.