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Asymptotic behavior of Hill's estimate and applications

Published online by Cambridge University Press:  24 August 2016

Gane Samb Lo
Gane Samb Lo*
Affiliation:
Université Paris VI
*
∗Postal address: 81 Résidence d'Athis, 26 Rue de la Plaine Basse, 91200 Athis-Mons, France.
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Abstract

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

Keywords

ORDER STATISTICSDOMAIN OF ATTRACTIONFUNCTIONS SLOWLY VARYING AT ZERONORMING CONSTANTSCENTRAL LIMIT THEOREMWEAK LAW OF LARGE NUMBERS
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 4 , December 1986 , pp. 922 - 936
Copyright
Copyright © Applied Probability Trust 1986 

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