Multivariate extremes over a random number of observations

Abstract

AbstractThe 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. Using 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.