Information criteria for the number of directions of extremes in high-dimensional data

TL;DR

This paper develops and compares information criteria, including BIC and QAIC, for determining the number of extreme directions in high-dimensional multivariate extreme value analysis, addressing the challenge of selecting the number of observations for estimation.

Contribution

It introduces new information criteria based on sparse regular variation for identifying extreme directions and proposes a two-step procedure for selecting the number of directions and observations.

Findings

BIC and QAIC are consistent criteria for extreme directions.

MSEIC and AIC are inconsistent for this purpose.

Simulation and wind speed data demonstrate the criteria's effectiveness.

Abstract

In multivariate extreme value analysis, the estimation of the dependence structure in extremes is demanding, especially in the context of high-dimensional data. Therefore, a common approach is to reduce the model dimension by considering only the directions in which extreme values occur. In this paper, we use the concept of sparse regular variation recently introduced by Meyer and Wintenberger (2021) to derive information criteria for the number of directions in which extreme events occur, such as a Bayesian information criterion (BIC), a mean-squared error-based information criterion (MSEIC), and a quasi-Akaike information criterion (QAIC) based on the Gaussian likelihood function. As is typical in extreme value analysis, a challenging task is the choice of the number $k_n$ of observations used for the estimation. Therefore, for all information criteria, we present a two-step procedure to estimate both the number of directions of extremes and an optimal choice of $k_n$. We prove that the AIC of Meyer and Wintenberger (2023) and the MSEIC are inconsistent information criteria for the number of extreme directions whereas the BIC and the QAIC are consistent information criteria. Finally, the performance of the different information criteria is compared in a simulation study and applied on wind speed data.