On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions

TL;DR

This paper investigates the bias in likelihood inference for multivariate max-stable distributions when using occurrence times of maxima, revealing conditions for unbiased inference and proposing a bias reduction method.

Contribution

It clarifies the bias phenomenon in high-dimensional max-stable inference and introduces a bias reduction technique applicable to models like the logistic extreme value distribution.

Findings

Bias occurs when dimension exceeds the square root of the number of maxima vectors.

Unbiased inference is feasible in moderate dimensions under certain conditions.

A practical bias reduction method is demonstrated on the logistic model.

Abstract

Full likelihood-based inference for high-dimensional multivariate extreme value distributions, or max-stable processes, is feasible when incorporating occurrence times of the maxima; without this information, $d$-dimensional likelihood inference is usually precluded due to the large number of terms in the likelihood. However, some studies have noted bias when performing high-dimensional inference that incorporates such event information, particularly when dependence is weak. We elucidate this phenomenon, showing that for unbiased inference in moderate dimensions, dimension $d$ should be of a magnitude smaller than the square root of the number of vectors over which one takes the componentwise maximum. A bias reduction technique is suggested and illustrated on the extreme value logistic model.