Does bias reduction with external estimator of second order parameter work for endpoint

Abstract Bias reduction estimation for tail index has been studied in the literature. One method is to reduce bias with an external estimator of the second order regular variation parameter; see Gomes and Martins [2002. Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31]. It is known that negative extreme value index implies that the underlying distribution has a finite right endpoint. As far as we know, there exists no bias reduction estimator for the endpoint of a distribution. In this paper, we study the bias reduction method with an external estimator of the second order parameter for both the negative extreme value index and endpoint simultaneously. Surprisingly, we find that this bias reduction method for negative extreme value index requires a larger order of sample fraction than that for positive extreme value index. This finding implies that this bias reduction method for endpoint is less attractive than that for positive extreme value index. Nevertheless, our simulation study prefers the proposed bias reduction estimators to the biased estimators in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568].

[1]  Liang Peng,et al.  Asymptotically unbiased estimators for the extreme-value index , 1998 .

[2]  Liang Peng,et al.  Estimating the First‐ and Second‐Order Parameters of a Heavy‐Tailed Distribution , 2004 .

[3]  L. Haan,et al.  On optimising the estimation of high quantiles of a probability distribution , 2003 .

[4]  Michael Woodroofe,et al.  Maximum Likelihood Estimation of Translation Parameter of Truncated Distribution II , 1974 .

[5]  Holger Drees,et al.  Refined pickands estimators wtth bias correction , 1996 .

[6]  L. Haan,et al.  A moment estimator for the index of an extreme-value distribution , 1989 .

[7]  Laurens de Haan,et al.  Third order extended regular variation , 2006 .

[8]  Peter Hall,et al.  On Estimating the Endpoint of a Distribution , 1982 .

[9]  Frederico Caeiro,et al.  Bias reduction of a tail index estimator through an external estimation of the second-order parameter , 2004 .

[10]  P. Hall,et al.  Estimating a tail exponent by modelling departure from a Pareto distribution , 1999 .

[11]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[12]  Jan Beirlant,et al.  On Exponential Representations of Log-Spacings of Extreme Order Statistics , 2002 .

[13]  K. Athreya,et al.  Confidence intervals for endpoints of a c.d.f. via bootstrap , 1997 .

[14]  M. Gomes,et al.  Bias reduction and explicit semi-parametric estimation of the tail index , 2004 .

[15]  P. Hall,et al.  Estimating the end-point of a probability distribution using minimum-distance methods , 1999 .

[16]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[17]  Holger Drees,et al.  On Smooth Statistical Tail Functionals , 1998 .

[18]  W. Loh,et al.  Estimating an Endpoint of a Distribution with Resampling Methods , 1984 .

[19]  M. J. Martins,et al.  “Asymptotically Unbiased” Estimators of the Tail Index Based on External Estimation of the Second Order Parameter , 2002 .

[20]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[21]  Michael Falk,et al.  Some Best Parameter Estimates for Distributions with Finite Endpoint , 1995 .