Lehmer's mean-of-order-p extreme value index estimation: a simulation study and applications

Abstract

The main objective of extreme value theory is essentially the estimation of quantities related to extreme events. One of its main issues has been the estimation of the extreme value index (EVI), a parameter directly related to the tail weight of the distribution. Here we deal with the semi-parametric estimation of the EVI, for heavy tails. A recent class of EVI-estimators, based on the Lehmer's mean-of-order p (L_p), which generalizes the arithmetic mean, is considered. An asymptotic comparison at optimal levels performed in previous works has revealed the competitiveness of this class of EVI-estimators. A large-scale Monte-Carlo simulation study for finite simulated samples has been here performed, showing the behaviour of L_p, as a function of p. A bootstrap adaptive choice of (k, p), where k is the number of upper order statistics used in the estimation, and a second algorithm based on a stability criterion are computationally studied and applied to simulated and real data.

Keywords:

• Generalized means
• heavy tails
• Monte-Carlo simulation
• semi-parametric estimation
• statistics of extremes

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research partially supported by National Funds through FCT – Fundação para a Ciência e a Tecnologia, projects UID/MAT/00006/2019 (CEA/UL), PEst-OE/MAT/UI0297/2019 (CMA/UNL), and by COST Action IC1408. We also thank our colleagues Ana Sá and José Miguel Cardoso Pereira for providing us with the burned areas data set. We further acknowledge the valuable suggestions from the referees and the Associate Editor.