On Second Order Conditions in the Multivariate Block Maxima and Peak over Threshold Method

TL;DR

This paper explores the relationship between second order conditions for multivariate block maxima and peak-over-threshold methods, revealing how they influence convergence rates and applying the theory to Archimax copulas.

Contribution

It establishes the connection between second order conditions for BM and POT methods in multivariate cases and analyzes their implications for different models.

Findings

Second order conditions for BM and POT often imply each other with different parameters.

Depending on the data, either BM or POT can achieve faster convergence.

In Archimax copulas, the second order parameter varies between methods.

Abstract

Second order conditions provide a natural framework for establishing asymptotic results about estimators for tail related quantities. Such conditions are typically tailored to the estimation principle at hand, and may be vastly different for estimators based on the block maxima (BM) method or the peak-over-threshold (POT) approach. In this paper we provide details on the relationship between typical second order conditions for BM and POT methods in the multivariate case. We show that the two conditions typically imply each other, but with a possibly different second order parameter. The latter implies that, depending on the data generating process, one of the two methods can attain faster convergence rates than the other. The class of multivariate Archimax copulas is examined in detail; we find that this class contains models for which the second order parameter is smaller for the BM method and vice versa. The theory is illustrated by a small simulation study.