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 distinguish 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 (MO\(_p\)) and build associated MO\(_p\) EVI-estimators. The normal asymptotic behaviour of MO\(_p\) EVI-estimators has already been obtained for \(p<1/(2\xi )\), with consistency achieved for \(p<1/\xi \), where \(\xi \) denotes the EVI. We shall now consider the non-regular case, \(p\ge 1/(2\xi )\), 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.