Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-26T15:20:47.284Z Has data issue: false hasContentIssue false
  • English
  • Français

On record values and the exponent of a distribution with regularly varying upper tail

Part of: Parametric inference

Published online by Cambridge University Press:  14 July 2016

Mohamed Berred
Mohamed Berred*
Affiliation:
Université Paris-VI
*
∗Postal address: C.R.E.S.T.-E.N.S.A.E., 12 rue Boulitte, 75014 Paris, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.

Keywords

RECORDSEXTREME VALUESREGULAR VARIATIONPARAMETRIC ESTIMATIONTAIL ESTIMATION

MSC classification

Primary: 62F10: Point estimation
Secondary: 62F12: Asymptotic properties of estimators
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 575 - 586
Copyright
Copyright © Applied Probability Trust 1992 

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.)

Footnotes

This work was supported by C.R.E.S.T.-E.N.S.A.E.

References

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Cambridge University Press.Google Scholar
Csörgő, S., Deheuvels, P. and Mason, D. M. (1985) Kernel estimate of the tail index of a distribution. Ann. Statist. 13, 1050–1077.CrossRefGoogle Scholar
Deheuvels, P. and Mason, D. M. (1988) The asymptotic behavior of sums of exponential extreme values. Bull. Sci. Math. (2) 112, 211–233.Google Scholar
Deheuvels, P., Haeusler, E. and Mason, D. M. (1988) Almost sure convergence of the Hill estimator. Math. Proc. Camb. Phil. Soc. 104, 371–381.CrossRefGoogle Scholar
Deheuvels, P., Haeusler, E. and Mason, D. M. (1990) On the almost sure behavior of sums of extreme values from a distribution in the domain of attraction of a Gumbel law. Bull. Sci. Math. (2) 114, 61–95.Google Scholar
Goldie, C. M. and Smith, R. L. (1987) Slow variation with remainder: Theory and applications. Quart. J. Math. Oxford (2) 38, 45–71.CrossRefGoogle Scholar
De Haan, L. and Resnick, S. I. (1980) A simple asymptotic estimate for the index of a stable distribution. J. R. Statist. Soc. B42, 83–87.Google Scholar
Haeusler, E. and Teugels, J. L. (1985) On asymptotic normality of Hill estimator for the exponent of a regular variation. Ann. Statist. 13, 743–757.Google Scholar
Hall, P. (1982) On some simple estimates of an exponent of regular variation. J. R. Statist. Soc. B44, 37–42.Google Scholar
Hanson, D. L. and Russo, R. P. (1981) On the law of large numbers. Ann. Prob. 9, 513–519.CrossRefGoogle Scholar
Hanson, D. L. and Russo, R. P. (1983) Some results on increments of the wiener process with applications to lag sums of i.i.d. random variables. Ann. Prob. 11, 609–623.Google Scholar
Hill, B. M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.CrossRefGoogle Scholar
Mason, D. M. (1982) Laws of large numbers for sums of extreme values. Ann. Prob. 10, 754–764.Google Scholar
Nevzorov, V. B. (1987) Records. Theory Prob. Appl. 32, 201–228.CrossRefGoogle Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New York.Google Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
Smith, R. L. (1982) Uniform rates of convergence in extreme value theory. Adv. Appl. Prob. 14, 600–622.Google Scholar
Teugels, J. L. (1981a) Limit theorems on order statistics. Ann. Prob. 9, 868–880.Google Scholar
Teugels, J. L. (1981b) Remarks on large claims. Bull. Inst. Internat. Statist. 49, 1490–1500.Google Scholar