Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology

References

• Beirlant , J. , Dierckx , G. , Goegebeur , Y. , Matthys , G. ( 1999 ). Tail index estimation and an exponential regression model . Extremes 2 : 177 – 200 .
   Google Scholar
• Beirlant , J. , Goegebeur , Y. , Segers , J. , Teugels , J. ( 2004 ). Statistics of Extremes. Theory and Applications . New York : Wiley .
   Google Scholar
• Beirlant , J. , Figueiredo , F. , Gomes , M. I. , Vandewalle , B. ( 2008 ). Improved reduced-bias tail index and quantile estimators . J. Statist. Plann. Infer. 138 ( 6 ): 1851 – 1870 .
   Google Scholar
• Caeiro , F. , Gomes , M. I. ( 2006 ). A new class of estimators of a “scale” second order parameter . Extremes 9 : 193 – 211 .
   Google Scholar
• Caeiro , F. , Gomes , M. I. ( 2008 ). Minumum-variance reduced-bias tail index and high quantile estimation . Revstat 6 ( 1 ): 1 – 20 .
   Web of Science ®Google Scholar
• Caeiro , F. , Gomes , M. I. , Pestana , D. D. ( 2005 ). Direct reduction of bias of the classical Hill estimator . Revstat 3 ( 2 ): 111 – 136 .
   Google Scholar
• Caeiro , F. , Gomes , M. I. , Henriques-Rodrigues , L. ( 2009 ). Reduced-bias tail index estimators under a third order framework . Commun. Statist. Theor. Meth. 38 ( 7 ): 1019 – 1040 .
   Web of Science ®Google Scholar
• Ciuperca , G. , Mercadier , C. ( 2010 ). Semi-parametric estimation for heavy tailed distributions . Extremes 13 ( 1 ): 55 – 87 .
   Web of Science ®Google Scholar
• Danielsson , J. , de Haan , L. , Peng , L. , de Vries , C. G. ( 2001 ). Using a bootstrap method to choose the sample fraction in the tail index estimation . J. Multivariate Anal. 76 : 226 – 248 .
   Web of Science ®Google Scholar
• Draisma , G. , de Haan , L. , Peng , L. , Pereira , T. ( 1999 ). A bootstrap-based method to achieve optimality in estimating the extreme value index . Extremes 2 ( 4 ): 367 – 404 .
   Google Scholar
• Feuerverger , A. , Hall , P. ( 1999 ). Estimating a tail exponent by modelling departure from a Pareto distribution . Ann. Statist. 27 : 760 – 781 .
   Web of Science ®Google Scholar
• Fraga Alves , M. I. , Gomes M. I., de Haan , L. (2003). A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60(2):194–213.
   Google Scholar
• Goegebeur , Y. , Beirlant , J. , de Wet , T. ( 2008 ). Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation . Revstat. 6 ( 1 ): 51 – 69 .
   Web of Science ®Google Scholar
• Goegebeur , Y. , Beirlant , J. , de Wet , T. ( 2010 ). Kernel estimators for the second order parameter in extreme value statistics . J. Statist. Plann. Infer. 140 ( 9 ): 2632 – 2652 .
   Web of Science ®Google Scholar
• Gomes , M. I. , Figueiredo , F. ( 2006 ). Bias reduction in risk modelling: Semi-parametric quantile estimation . Test 15 ( 2 ): 375 – 396 .
   Web of Science ®Google Scholar
• Gomes , M. I. , Martins , M. J. ( 2002 ). “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter . Extremes 5 ( 1 ): 5 – 31 .
   Google Scholar
• Gomes , M. I. , Oliveira , O. ( 2001 ). The bootstrap methodology in Statistics of Extremes: Choice of the optimal sample fraction . Extremes 4 ( 4 ): 331 – 358 .
   Google Scholar
• Gomes , M. I. , Pestana , D. D. ( 2007a ). A simple second order reduced-bias’ tail index estimator . J. Statist. Computat. Simul. 77 ( 6 ): 487 – 504 .
   Web of Science ®Google Scholar
• Gomes , M. I. , Pestana , D. D. ( 2007b ). A sturdy reduced-bias extreme quantile (VaR) estimator . J. Amer. Statist. Assoc. 102 ( 477 ): 280 – 292 .
   Web of Science ®Google Scholar
• Gomes , M. I. , Martins , M. J. , Neves , M. M. ( 2000 ). Alternatives to a semi-parametric estimator of parameters of rare events—the Jackknife methodology . Extremes 3 ( 3 ): 207 – 229 .
   Google Scholar
• Gomes , M. I. , Martins , M. J. , Neves , M. M. ( 2007 ). Improving second order reduced bias extreme value index estimation . Revstat 5 ( 2 ): 177 – 207 .
   Web of Science ®Google Scholar
• Gomes , M. I. , Canto e Castro , L. , Fraga Alves , M. I. , Pestana , D. ( 2008a ). Statistics of extremes for iid data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions . Extremes 11 ( 1 ): 3 – 34 .
   Web of Science ®Google Scholar
• Gomes , M. I. , de Haan , L. , Henriques-Rodrigues , L. ( 2008b ). Tail Index estimation for heavy-tailed models: Accommodation of bias in weighted log-excesses . J. Roy. Statist. Soc. B70 ( 1 ): 31 – 52 .
   Google Scholar
• Gomes , M. I. , Mendonca , S. , Pestana , D. D. ( 2009 ). Adaptive reduced-bias tail index and value-at-risk estimation . In: Sakalauskas , L. , Skiadas , C. , Zavadskas , E. K. , eds. Applied Stochastic Models and Data Analysis 2009 . IMI and VGTU Editions. Vilnius , Lithuania : Vilnius Gediminas Technical University , pp. 41 – 44 .
   Google Scholar
• Gomes , M. I. , Henriques-Rodrigues , L. , Pereira , H. , Pestana , D. ( 2010 ). Tail index and second order parameters’ semi-parametric estimation based on the log-excesses . J. Statist. Computat. Simul. 80 ( 6 ): 653 – 666 .
   Web of Science ®Google Scholar
• Hall , P. ( 1990 ). Using bootstrap to estimate mean squared error and selecting parameter in nonparametric problems . J. Multivariate Anal. 32 : 177 – 203 .
   Web of Science ®Google Scholar
• Hill , B. ( 1975 ). A simple general approach to inference about the tail of a distribution . Ann. Statist. 3 : 1163 – 1174 .
   Web of Science ®Google Scholar
• Li , D. , Peng , L. , Yang , J. ( 2010 ). Bias reduction for high quantiles . J. Statist. Plann. Infer. 140 : 2433 – 2441 .
   Web of Science ®Google Scholar
• Matthys , G. , Beirlant , J. ( 2003 ). Estimating the extreme value index and high quantiles with exponential regression models . Statistica Sinica 13 : 853 – 880 .
   Web of Science ®Google Scholar
• Matthys , G. , Delafosse , M. , Guillou , A. , Beirlant , J. ( 2004 ). Estimating catastrophic quantile levels for heavy-tailed distributions . Insur. Math. Econ. 34 : 517 – 537 .
   Web of Science ®Google Scholar
• Peng , L. ( 1998 ). Asymptotically unbiased estimator for the extreme-value index . Statist. Probab. Lett. 38 ( 2 ): 107 – 115 .
   Web of Science ®Google Scholar
• Reiss , R.-D. , Thomas , M. ( 2007 ). Statistical Analysis of Extreme Values, with Application to Insurance, Finance, Hydrology and Other Fields. , 3rd ed. Basel , Switzerland : Birkhäuser Verlag .
   Google Scholar
• Vandewalle , B. , Beirlant , J. ( 2006 ). On univariate extreme value statistics and the estimation of reinsurance premiums . Insur. Math. Econ. 38 ( 3 ): 444 – 459 .
   Google Scholar
• Weissman , I. ( 1978 ). Estimation of parameters and large quantilesbased on the k largest observations . J. Amer. Statist. Assoc. 73 : 812 – 815 .
   Web of Science ®Google Scholar