Non-regular Frameworks and the Mean-of-Order p Extreme Value Index Estimation

Abstract

Most of the estimators of parameters of rare and large events, among which we dis- tinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory, are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarithm of the power mean of order p = 0 of a certain set of statistics. Only for heavy tails, i.e. a positive EVI, quite common in many areas of application, and trying to improve the performance of the classical Hill EVI-estimators, instead of the aforementioned geometric mean, we can more generally consider the power mean of order-p (MOp) and build associated MOp EVI-estimators. The normal asymptotic behaviour of MOp EVI-estimators has already been obtained for p < 1/(2ξ), with consistency achieved for p < 1/ξ , where ξ denotes the EVI. We shall now consider the non-regular case, p ≥ 1/(2ξ ), a situation in which either normal or non-normal sum- stable laws can be obtained, together with the possibility of an ‘almost degenerate’ EVI-estimation.