Data Analysis and Related Applications, Volume 2

16A Generalized Mean Under a Non-Regular Framework and Extreme Value Index Estimation

The Hill estimator, one of the most popular extreme value index (EVI) estimators under a heavy right-tail framework, i.e. for a positive EVI, here denoted by ξ, is an average of the log-excesses. Consequently, it can be regarded as the logarithm of the geometric mean or mean of order p = 0 of an adequate set of systematic statistics. We can thus more generally consider any real p, the mean of order p (MO_p) of those same statistics and the associated MO_p EVI-estimators, also called harmonic moment EVI-estimators. The normal asymptotic behavior of these estimators has been obtained for p < 1/(2ξ), with consistency achieved for p < 1/ξ. The non-regular framework, i.e. the case p ≥ 1/(2ξ), will be now considered. Consistency is no longer achieved for p > 1/ξ, but an almost degenerate behavior appears for p = 1/ξ. The results are illustrated on the basis of large-scale simulation studies. An algorithm providing an almost degenerate MO_p EVI-estimation is suggested.

16.1. Introduction

Given X₁,…, X_n, a sample of size n of independent, identically distributed (IID), or possibly stationary weakly dependent random variables (RVs), with a cumulative distribution function (CDF) F, let us consider the notation X_1:n ≤ … ≤ X_n:n for the associated ascending order statistics (OSs). Let us further assume that there exist real constants a_n > 0 and b_n ∈ ℝ such that the linearly normalized maximum, (X_n:n − b