Several new tail index estimators

In the paper we propose some new class of functions which is used to construct tail index estimators. Functions from this new class is non-monotone in general, but presents a product of two monotone functions: the power function and the logarithmic function, which plays essential role in the classical Hill estimator. Introduced new estimators have better asymptotic performance comparing with the Hill estimator and other popular estimators over all range of the parameters present in the second order regular variation condition. Asymptotic normality of the introduced estimators is proved, and comparison (using asymptotic mean square error) with other estimators of the tail index is provided. Some preliminary simulation results are presented.

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

[2]  Milan Stehlík,et al.  On the favorable estimation for fitting heavy tailed data , 2010, Comput. Stat..

[3]  Jon Danielsson,et al.  The method of moments ratio estimator for the tail shape parameter , 1996 .

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

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

[6]  A. Dekkers,et al.  Optimal choice of sample fraction in extreme-value estimation , 1993 .

[7]  Vygantas Paulauskas,et al.  A New Estimator for a Tail Index , 2003 .

[8]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[9]  Jan Beran,et al.  The harmonic moment tail index estimator: asymptotic distribution and robustness , 2014 .

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

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

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

[13]  H. Drees.  A general class of estimators of the extreme value index , 1998 .

[14]  V. Paulauskas,et al.  On an improvement of Hill and some other estimators , 2013 .

[15]  Frederico Caeiro,et al.  A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator , 2009 .

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

[17]  Vygantas Paulauskas,et al.  Once more on comparison of tail index estimators , 2011, 1104.1242.

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

[19]  V. Paulauskas,et al.  Several modifications of DPR estimator of the tail index , 2011 .

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