Asymptotic normality of the likelihood moment estimators for a stationary linear process with heavy-tailed innovations

TL;DR

This paper establishes the asymptotic normality of likelihood moment estimators for parameters of the Generalized Pareto distribution when applied to data from a stationary linear process with heavy-tailed innovations, extending their theoretical understanding.

Contribution

It proves the bivariate asymptotic normality of likelihood moment estimators for heavy-tailed linear processes, including explicit conditions verification methods.

Findings

Likelihood moment estimators are asymptotically normal for heavy-tailed linear processes.

Provided explicit conditions for checking asymptotic normality using asymptotic expansions.

Extended the theoretical foundation for using these estimators in dependent heavy-tailed data.

Abstract

A variety of estimators for the parameters of the Generalized Pareto distribution, the approximating distribution for excesses over a high threshold, have been proposed, always assuming the underlying data to be independent. We recently proved that the likelihood moment estimators are consistent estimators for the parameters of the Generalized Pareto distribution for the case where the underlying data arises from a (stationary) linear process with heavy-tailed innovations. In this paper we derive the bivariate asymptotic normality under some additional assumptions and give an explicit example on how to check these conditions by using asymptotic expansions.