Hostname: page-component-5c6d5d7d68-7tdvq Total loading time: 0 Render date: 2024-08-16T00:57:19.625Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Nonparametric inference Limit theorems 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: pakes@maths.uwa.edu.au
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bingham, N. H., Goldie, C. and Teugels, J. F. (1987). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.Google Scholar
[2] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
[3] Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[4] Hashorva, E. and Hüsler, J. (1999). On the number of near-maxima. Preprint, Inst. Math. Statist., University of Bern.Google Scholar
[5] Li, Y. (1998). The Number of Near-Maxima of Random Variables. , University of Western Australia.Google Scholar
[6] Li, Y. (1999). A note on the number of records near the maximum. Statist. Prob. Lett. 43, 153–158.Google Scholar
[7] Li, Y. and Pakes, A. G. (2000). On the number of near-maximum insurance claims. To appear in Insurance: Math. Econom.Google Scholar
[8] Maller, R. A. and Resnick, S. I. (1984). Limiting behaviour of sums and the term of maximum modulus. Proc. London Math. Soc. 49, 385–422.Google Scholar
[9] Pakes, A. G. and Li, Y. (1998). Limit laws for the number of near maxima via the Poisson approximation. Statist. Prob. Lett. 40, 395–401.CrossRefGoogle Scholar
[10] Pakes, A. G. and Steutel, F. W. (1997). On the number of records near the maximum. Austral. J. Statist. 39(2), 179–192.CrossRefGoogle Scholar
[11] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
[12] Royden, H. L. (1988). Real Analysis, 3rd edn. Macmillan, New York.Google Scholar