Direct Reduction of Bias of the Classical Hill Estimator

Frederico Caeiro · M. Ivette Gomes · Dínis Pestana

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Probability and Risk Models Probability and Statistical Research Statistical Distribution Estimation and Applications

Abstract

In this paper we are interested in an adequate estimation of the dominant component of the bias of Hill’s estimator of a positive tail index γ, in order to remove it from the classical Hill estimator in different asymptotically equivalent ways. If the second order parameters in the bias are computed at an adequate level k1 of a larger order than that of the level k at which the Hill estimator is computed, there may be no change in the asymptotic variances of these reduced bias tail index estimators, which are kept equal to the asymptotic variance of the Hill estimator, i.e., equal to γ 2 . The asymptotic distributional properties of the proposed estimators of γ are derived and the estimators are compared not only asymptotically, but also for finite samples through Monte Carlo techniques.

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References

1982On Some Simple Estimates of an Exponent of Regular Variation 1999Tail Index Estimation and an Exponential Regression Model 1999Estimating a tail exponent by modelling departure from a Pareto distribution 1998Comparison of tail index estimators 2002“Asymptotically Unbiased” Estimators of the Tail Index Based on External Estimation of the Second Order Parameter

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