A new partially reduced-bias mean-of-order p class of extreme value index estimators

A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order- p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k , the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.

[1]  Jan Beirlant,et al.  Tail Index Estimation and an Exponential Regression Model , 1999 .

[2]  Piet Groeneboom,et al.  Kernel-type estimators for the extreme value index , 2003 .

[3]  James P. Hughes,et al.  Computing the observed information in the hidden Markov model using the EM algorithm , 1997 .

[4]  Laurens de Haan,et al.  On maximum likelihood estimation of the extreme value index , 2004, math/0407062.

[5]  Cécile Mercadier,et al.  Semi-parametric estimation for heavy tailed distributions , 2010 .

[6]  Jan Beirlant,et al.  Excess functions and estimation of the extreme-value index , 1996 .

[7]  M. Gomes,et al.  Statistics of extremes for IID data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions , 2008 .

[8]  M. Ivette Gomes,et al.  Reduced-Bias Tail Index Estimators Under a Third-Order Framework , 2009 .

[9]  Alan H. Welsh,et al.  Adaptive Estimates of Parameters of Regular Variation , 1985 .

[10]  M. Ivette Gomes,et al.  IMPROVING SECOND ORDER REDUCED BIAS EXTREME VALUE INDEX ESTIMATION , 2007 .

[11]  M. Ivette Gomes,et al.  A simple generalisation of the Hill estimator , 2013, Comput. Stat. Data Anal..

[12]  M. Ivette Gomes,et al.  Extreme Value Theory and Statistics of Univariate Extremes: A Review , 2015 .

[13]  M. Ivette Gomes,et al.  Peaks over random threshold methodology for tail index and high quantile estimation , 2006 .

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

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

[16]  Frederico Caeiro,et al.  A new class of estimators of a “scale” second order parameter , 2007 .

[17]  M. Ivette Gomes,et al.  The Bootstrap Methodology in Statistics of Extremes—Choice of the Optimal Sample Fraction , 2001 .

[18]  M. Ivette Gomes,et al.  A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator , 2007 .

[19]  George Kingsley Zipf,et al.  National Unity and Disunity , 1943 .

[20]  Liang Peng,et al.  Review of testing issues in extremes: in honor of Professor Laurens de Haan , 2008 .

[21]  M. Gomes,et al.  Generalized Jackknife-Based Estimators for Univariate Extreme-Value Modeling , 2013 .

[22]  M. Ivette Gomes,et al.  Mixed moment estimator and location invariant alternatives , 2009 .

[23]  Hogeschool-Universiteit Brussel,et al.  GENERALIZED SUM PLOTS , 2011 .

[24]  M. Ivette Gomes,et al.  DIRECT REDUCTION OF BIAS OF THE CLASSI- CAL HILL ESTIMATOR ⁄ , 2005 .

[25]  Liang Peng,et al.  Comparison of tail index estimators , 1998 .

[26]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[27]  Anthony C. Davison,et al.  Modelling Excesses over High Thresholds, with an Application , 1984 .

[28]  L. Haan,et al.  Extreme value theory : an introduction , 2006 .

[29]  M. Ivette Gomes,et al.  A new class of semi-parametric estimators of the second order parameter. , 2003 .

[30]  Laurens de Haan,et al.  Slow Variation and Characterization of Domains of Attraction , 1984 .

[31]  R. Reiss,et al.  Statistical Analysis of Extreme Values-with applications to insurance , 1997 .

[32]  L. Haan,et al.  Bias correction in extreme value statistics with index around zero , 2013 .

[33]  Jan Beirlant,et al.  Kernel estimators for the second order parameter in extreme value statistics , 2010 .

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

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

[36]  Jan Beirlant,et al.  LINKING PARETO-TAIL KERNEL GOODNESS-OF-FIT STATISTICS WITH TAIL INDEX AT OPTIMAL THRESHOLD AND SECOND ORDER ESTIMATION , 2008 .

[37]  Frederico Caeiro,et al.  Revisiting the Maximum Likelihood Estimation of a Positive Extreme Value Index , 2015 .

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

[39]  Paul Deheuvels,et al.  Kernel Estimates of the Tail Index of a Distribution , 1985 .

[40]  S. Girard,et al.  On the estimation of the second order parameter for heavy-tailed distributions , 2013 .

[41]  M. Neves,et al.  Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology* , 2000 .

[42]  George Michailidis,et al.  A Diagnostic Plot for Estimating the Tail Index of a Distribution , 2004 .

[43]  M. Gomes,et al.  Generalizations of the Hill estimator – asymptotic versus finite sample behaviour☆ , 2001 .

[44]  E. J. Gumbel,et al.  Statistics of Extremes. , 1960 .

[45]  Frederico Caeiro,et al.  Asymptotic comparison at optimal levels of reduced‐bias extreme value index estimators , 2011 .

[46]  J. Teugels,et al.  Statistics of Extremes , 2004 .

[47]  P. Hall.  On Some Simple Estimates of an Exponent of Regular Variation , 1982 .

[48]  M. Ivette Gomes,et al.  New Reduced-bias Estimators of a Positive Extreme Value Index , 2016, Commun. Stat. Simul. Comput..

[49]  J. Geluk,et al.  Regular variation, extensions and Tauberian theorems , 1987 .

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

[51]  Frederico Caeiro,et al.  A Semi-parametric Estimator of a Shape Second-Order Parameter , 2014 .

[52]  M. Ivette Gomes,et al.  Tail index estimation for heavy‐tailed models: accommodation of bias in weighted log‐excesses , 2007 .

[53]  Modeling Extreme Events: Sample Fraction Adaptive Choice in Parameter Estimation , 2015 .

[54]  M. Ivette Gomes,et al.  Adaptive PORT–MVRB estimation: an empirical comparison of two heuristic algorithms , 2013 .

[55]  Michael Thomas,et al.  Statistical Analysis of Extreme Values , 2008 .

[56]  Jan Beirlant,et al.  Estimation of the extreme-value index and generalized quantile plots , 2005 .