Semi-parametric estimation for heavy tailed distributions

In this paper, we generalize several studies in the area of extreme value theory for the estimation of the extreme value index and the second order parameter. Weak consistency and asymptotic normality are proven under classical assumptions. Some numerical simulations and computations are also performed to illustrate the finite-sample and the limiting behavior of the estimators.

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

[2]  S. Hubbert.  Extreme Value Theory , 2019, Handbook of Heavy-Tailed Distributions in Asset Management and Risk Management.

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

[4]  J. Hüsler,et al.  Weighted least squares estimation of the extreme value index , 2006 .

[5]  L. Haan,et al.  Extreme value theory , 2006 .

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

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

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

[9]  I. Weissman.  Estimation of Parameters and Large Quantiles Based on the k Largest Observations , 1978 .

[10]  Joseph L. Gastwirth,et al.  Asymptotic Distribution of Linear Combinations of Functions of Order Statistics with Applications to Estimation , 1967 .

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

[12]  J. Doob.  Stochastic processes , 1953 .

[13]  A. Guillou,et al.  A new extreme quantile estimator for heavy-tailed distributions , 2004 .

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

[15]  Ishay Weissman,et al.  Estimation of parameters and large quantiles based on the K largest observations , 1978, Advances in Applied Probability.

[16]  S. Resnick,et al.  The qq-estimator and heavy tails , 1996 .

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

[18]  M. Gomes,et al.  Reduced‐bias tail index estimation and the jackknife methodology , 2007 .

[19]  Liang Peng,et al.  Semi-parametric Estimation of the Second Order Parameter in Statistics of Extremes , 2002 .

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

[21]  Josef Steinebach,et al.  ON LEAST SQUARES ESTIMATES OF AN EXPONENTIAL TAIL COEFFICIENT , 1996 .

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

[23]  Edgar Kaufmann,et al.  Selecting the optimal sample fraction in univariate extreme value estimation , 1998 .