On the block maxima method in extreme value theory: PWM estimators

Abstract

In extreme value theory, there are two fundamental approaches, both widely used: the block maxima (BM) method and the peaksover-threshold (POT) method. Whereas much theoretical research has gone into the POT method, the BM method has not been studied thoroughly. The present paper aims at providing conditions under which the BM method can be justified. We also provide a theoretical comparative study of the methods, which is in general consistent with the vast literature on comparing the methods all based on simulated data and fully parametric models. The results indicate that the BM method is a rather efficient method under usual practical conditions.In this paper, we restrict attention to the i.i.d. case and focus on the probability weighted moment (PWM) estimators of Hosking, Wallis and Wood [Technometrics (1985) 27 251-261].1. Introduction. The block maxima (BM) approach in extreme value theory (EVT), consists of dividing the observation period into nonoverlapping periods of equal size and restricts attention to the maximum observation in each period [see, e.g., Gumbel (1958)]. The new observations thus created follow-under domain of attraction conditions, cf. (2) belowapproximately an extreme value distribution, G γ for some real γ. Parametric statistical methods for the extreme value distributions are then applied to those observations. In the peaks-over-threshold (POT) approach in EVT, one selects those of the initial observations that exceed a certain high threshold. The probability distribution of those selected observations is approximately a generalized Pareto distribution Pickands (1975).

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