Smoothing the Moment Estimator of the Extreme Value Parameter

Let {Xn be a sequence of i.i.d. random variables whose common distribution F belongs to the domain of attraction of an extreme value law. A semi-parametric estimator of the extreme value parameter is the Dekkers, Einmahl and de Haan [8] moment estimator. Practical use of this estimator requires the problematic choice of a number k=k(n) of upper order statistics and there are few reliable guidelines for this choice. An averaging or smoothing technique is proposed for this estimator yielding a less volatile function of k which in practice aids estimation.

[1]  S. Resnick,et al.  Consistency of Hill's estimator for dependent data , 1995, Journal of Applied Probability.

[2]  E. Haeusler,et al.  On Asymptotic Normality of Hill's Estimator for the Exponent of Regular Variation , 1985 .

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

[4]  G. Draisma,et al.  An estimator for the extreme-value index , 1996 .

[5]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[6]  Sidney I. Resnick,et al.  How to make a Hill Plot , 2000 .

[7]  Richard L. Smith.  Estimating tails of probability distributions , 1987 .

[8]  L. Haan,et al.  On the Estimation of the Extreme-Value Index and Large Quantile Estimation , 1989 .

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

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

[11]  Sidney I. Resnick,et al.  On asymptotic normality of the hill estimator , 1998 .

[12]  Tailen Hsing,et al.  On Tail Index Estimation Using Dependent Data , 1991 .

[13]  D. Mason,et al.  Intermediate- and extreme-sum processes , 1992 .

[14]  U. Stadtmüller,et al.  Generalized regular variation of second order , 1996, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

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

[16]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

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

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

[19]  L. Haan,et al.  Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation , 2000 .

[20]  Charles M. Goldie,et al.  SLOW VARIATION WITH REMAINDER: THEORY AND APPLICATIONS , 1987 .

[21]  Malcolm R Leadbetter,et al.  Tail and Quantile Estimation for Strongly Mixing Stationary Sequences , 1990 .

[22]  Sidney Resnick,et al.  Smoothing the Hill Estimator , 1997, Advances in Applied Probability.

[23]  D. Mason.  Laws of Large Numbers for Sums of Extreme Values , 1982 .

[24]  Sidney I. Resnick,et al.  Estimating the limit distribution of multivariate extremes , 1993 .

[25]  W. Vervaat.  Functional central limit theorems for processes with positive drift and their inverses , 1972 .

[26]  N. Bingham,et al.  Extensions of Regular Variation, I: Uniformity and Quatifiers , 1982 .

[27]  J. Hosking,et al.  Parameter and quantile estimation for the generalized pareto distribution , 1987 .

[28]  D. Mason,et al.  A strong invariance theorem for the tail empirical process , 1988 .

[29]  Sidney I. Resnick,et al.  Tail estimates motivated by extreme-value theory , 1984, Advances in Applied Probability.

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

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