Generalized Kernel Estimators for the Weibull-Tail Coefficient

We introduce new families of estimators for the Weibull-tail coefficient, obtained from a weighted sum of a power transformation of excesses over a high random threshold. Asymptotic normality of the estimators is proven for an intermediate sequence of upper order statistics, and under classical regularity conditions for L-statistics and a second-order condition on the tail behavior of the underlying distribution. The small sample performance of two specific examples of kernel functions is evaluated in a simulation study.

[1]  J. Diebolt,et al.  Bias-reduced estimators of the Weibull tail-coefficient , 2008, 1103.6172.

[2]  Laurent Gardes,et al.  Comparison of Weibull tail-coefficient estimators , 2011, 1104.0764.

[3]  M. Broniatowski.  On the estimation of the Weibull tail coefficient , 1993 .

[4]  A. Rényi.  On the theory of order statistics , 1953 .

[5]  Laurent Gardes,et al.  Estimating Extreme Quantiles of Weibull Tail Distributions , 2005 .

[6]  M. Meerschaert.  Regular Variation in R k , 1988 .

[7]  S. Girard.  A Hill Type Estimator of the Weibull Tail-Coefficient , 2004 .

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

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

[10]  Laurent Gardes,et al.  Estimation of the Weibull tail-coefficient with linear combination of upper order statistics , 2008, 1103.5894.

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

[12]  Armelle Guillou,et al.  Goodness-of-fit testing for Weibull-type behavior , 2010 .

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

[14]  D. Mason.  Asymptotic Normality of Linear Combinations of Order Statistics with a Smooth Score Function , 1981 .

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

[16]  J. Teugels,et al.  The mean residual life function at great age: Applications to tail estimation , 1995 .

[17]  David J. Edwards,et al.  Mean Residual Life , 2011, International Encyclopedia of Statistical Science.