Flexible Extreme Value Inference And Hill Plots For A Small, Mid And Large Samples

TL;DR

This paper introduces a flexible approach to extreme value inference and Hill plots that avoids complex second-order regularity conditions, making tail estimation more practical for small to large samples, with applications to ecological data.

Contribution

It provides a novel methodology for tail estimation that relies only on Karamata's representation, bypassing second-order conditions and enhancing applicability to real data.

Findings

New methodology for tail estimation without second-order conditions

Introduction of alternative Hill plots for better data visualization

Application to ecological snow extreme data

Abstract

Asymptotic normality of extreme value tail estimators received much attention in the literature, giving rise to increasingly complicated 2nd order regularity conditions. However, such conditions are really difficult to be checked for real data. Especially it is difficult or impossible to check such conditions using small samples. Beside that most of those conditions suffer from the drawback of a potentially singular integral representations. However, we can have various orders of approximation by normal distributions, e.g. Berry-Essen Types and Edgeworth types. In this paper we indicate that for Berry-Essen Types of normal approximation and related asymptotic normality of generalized Hill estimators, we do not necessarily need 2nd order regularity conditions and we can apply only Karamata's representation for regularly varying tails. 2nd order regularity conditions however better relates to Edgeworth types of normal approximations, albeit requiring larger data samples for their proper check. Finally both expansions are prone for bootstrap and other subsampling techniques. All existing results indicate that proper representation of tail behavior play a special and somewhat intriguing role in that context. We dispel that widespread opinion by providing a full characterization and representation, in a general regular variation context, of the integral singularity phenomenon, highlighting its relation to an asymptotical normality of the Generalized Hill estimator without the 2nd order condition. Thus application of this new methodology is simple and much more flexible, optimal for real data sets. Alternative and powerful versions of the Hill plot are also introduced and illustrated on ecological data of snow extremes from Slovakia.