Multivariate Extremes Over a Random Number of Observations

TL;DR

This paper extends multivariate extreme-value theory to aggregated data from a random number of observations, introducing new max-stable distributions and a semiparametric estimator for extremal dependence.

Contribution

It develops a limit theorem for aggregated data, introduces a new family of max-stable distributions, and proposes a semiparametric estimator for extremal dependence.

Findings

Derived a limit theorem for aggregated maxima with a random sample size.

Introduced a new family of max-stable distributions for aggregated data.

Demonstrated the estimator's performance through simulations.

Abstract

The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. By means of an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study.