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The number and sum of near-maxima for thin-tailed populations

Part of: Limit theorems Nonparametric inference Stochastic processes Real functions

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
The University of Western Australia
*
∗ Postal address: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Email address: [email protected]
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Abstract

A near-maximum is an observation which falls within a distance a of the maximum observation in an i.i.d. sample of size n. The asymptotic behaviour of the number Kn(a) of near-maxima is known for the cases where the right extremity of the population distribution function is finite, and where it is infinite and the right hand tail is exponentially small, or fatter than exponential. This paper completes the picture for thin tails, i.e., tails which decay faster than exponential. Limit theorems are derived and used to find the large-sample behaviour of the sum of near-maxima.

Keywords

Sample maximanear-maximaorder statisticslimit theoremsPoisson approximationregular variationtrimmed means

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems 26A12: Rate of growth of functions, orders of infinity, slowly varying functions 62G30: Order statistics; empirical distribution functions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1100 - 1116
Copyright
Copyright © Applied Probability Trust 2000 

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