Generalized R\'enyi statistics

Abstract

In R\'enyi's representation for exponential order statistics, we replace the iid exponential sequence with any iid sequence, and call the resulting order statistic generalized R\'enyi statistic. We prove that by randomly reordering the variables in the generalized R\'enyi statistic, we obtain in the limit a sequence of iid exponentials. This result allows us to propose a new model for heavy-tailed data. Although the new model is very close to the classical iid framework, we establish that the Hill estimator is weakly consistent and asymptotically normal without any further assumptions on the underlying distribution or on the number of upper order statistics used in the estimator.