Estimating a tail exponent by modelling departure from a Pareto distribution. (English) Zbl 0942.62059

Feuerverger, Andrey; Hall, Peter

Ann. Stat. 27, No. 2, 760-781 (1999).

Summary: We suggest two semiparametric methods for accommodating departures from a Pareto model when estimating a tail exponent by fitting the model to extreme-value data. The methods are based on approximate likelihood and on least squares, respectively. The latter is somewhat simpler to use and more robust against departures from classical extreme-value approximations, but produces estimators with approximately 64% greater variance when conventional extreme-value approximations are appropriate. Relative to the conventional assumption that the sampling population has exactly a Pareto distribution beyond a threshold, our methods reduce bias by an order of magnitude without inflating the order of variance. They are motivated by data on extrema of community sizes and are illustrated by an application in that context.

MSC:

62G32	Statistics of extreme values; tail inference
62G30	Order statistics; empirical distribution functions

Keywords:

bias reduction; extreme-value theory; log-spacings; maximum likelihood; order statistics; peaks-over-threshold; regression; regular variation; spacings Zipf’s law

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