A location-invariant probability weighted moment estimation of the Extreme Value Index

Abstract

The peaks over random threshold (PORT) methodology and the Pareto probability weighted moments (PPWM) of the largest observations are used to build a class of location-invariant estimators of the Extreme Value Index (EVI), the primary parameter in statistics of extremes. The asymptotic behaviour of such a class of EVI-estimators, the so-called PORT-PPWM EVI-estimators, is derived, and an alternative class of location-invariant EVI-estimators, the generalized Pareto probability weighted moments (GPPWM) EVI-estimators is considered as an alternative. These two classes of estimators, the PORT-PPWM and the GPPWM, jointly with the classical Hill EVI-estimator and a recent class of minimum-variance reduced-bias estimators are compared for finite samples, through a large-scale Monte-Carlo simulation study. An adaptive choice of the tuning parameters under play is put forward and applied to simulated and real data sets.

Keywords:

• adaptive semi-parametric estimation
• asymptotic properties
• bootstrap methodology
• Extreme Value Index
• heavy tails
• location/scale-invariant estimation
• Monte-Carlo simulation
• statistics of extremes

2000 AMS Subject Classifications:

• Primary: 62G32
• 62E20
• Secondary: 65C05

Acknowledgements

Research partially supported by National Funds through FCT – Fundação para a Ciência e a Tecnologia, projects PEst-OE/MAT/UI0006/2014 (CEAUL), PEst-OE/MAT/UI0297/2014 (CMA/UNL), EXTREMA project, PTDC/FEDER and grant SFRH/BPD/77319/2011.