Estimating the First‐ and Second‐Order Parameters of a Heavy‐Tailed Distribution

Summary This paper suggests censored maximum likelihood estimators for the first- and secondorder parameters of a heavy-tailed distribution by incorporating the second-order regular variation into the censored likelihood function. This approach is different from the biasreduced maximum likelihood method proposed by Feuerverger and Hall in 1999. The paper derives the joint asymptotic limit for the first- and second-order parameters under a weaker assumption. The paper also demonstrates through a simulation study that the suggested estimator for the first-order parameter is better than the estimator proposed by Feuerverger and Hall although these two estimators have the same asymptotic variances.

[1]  Jón Dańıelsson,et al.  The Cost of Conservatism: , 2000 .

[2]  László Viharos,et al.  Asymptotic normality of least-squares estimators of tail indices , 1997 .

[3]  Armelle Guillou,et al.  A diagnostic for selecting the threshold in extreme value analysis , 2001 .

[4]  Peter Hall,et al.  Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems , 1990 .

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

[6]  Gennady Samorodnitsky,et al.  LIVING ON THE EDGE , 2005 .

[7]  P. Hall,et al.  Estimating a tail exponent by modelling departure from a Pareto distribution , 1999 .

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

[9]  Jan Beirlant,et al.  Tail Index Estimation and an Exponential Regression Model , 1999 .

[10]  J. Teugels,et al.  Tail Index Estimation, Pareto Quantile Plots, and Regression Diagnostics , 1996 .

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

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

[13]  Peter Hall,et al.  On the estimation of extreme tail probabilities , 1997 .

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

[15]  Peter Hall,et al.  On Estimating the Endpoint of a Distribution , 1982 .

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

[17]  A. McNeil.  Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory , 1997, ASTIN Bulletin.

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

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

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

[21]  Jón Dańıelsson,et al.  Tail Index and Quantile Estimation with Very High Frequency Data , 1997 .

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

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

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