Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T03:19:37.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating probabilities for normal extremes

Published online by Cambridge University Press:  01 July 2016

Peter Hall
Peter Hall*
Affiliation:
The Australian National University
*
∗Postal address: Department of Statistics, SGS, The Australian National University, P.O. Box 4, Canberra, A.C.T. 2600, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Xnn denote the largest of n independent N(0, 1) variables. Several methods of estimating P(Xnn ≦ x) are considered. It is shown that X2nn, when normalized in an optimal way, converges to the extreme value distribution at a rate of only 1/(log n)2, and that if 0 < t ≠ 2 then |Xnn|t converges at a rate of 1/log n. Therefore it is not feasible to use the extreme value distribution to estimate probabilities for normal extremes unless the sample size is extremely large. An alternative approach is presented, which gives very good estimates of P(Xnn ≦ x) for n ≧ 10. The case of rth extremes is also considered.

Keywords

NORMAL DISTRIBUTIONEXTREME VALUEEXTREME VALUE DISTRIBUTIONRATE OF CONVERGENCEESTIMATENON-UNIFORM ESTIMATE
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 491 - 500
Copyright
Copyright © Applied Probability Trust 1980 

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

Anderson, C. W. (1976) Extreme value theory and its approximations. In Proc. Symp. Reliability Technology, Bradford. U.K. Atomic Energy Authority.Google Scholar
Dronkers, J. J. (1958) Approximate formulae for the statistical distributions of extreme values. Biometrika 45, 447–470.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Govindarajulu, Z. and Hubacker, N. W. (1964) Percentiles of order statistics in samples from uniform, normal, chi square (1 d.f.) and Weibull populations. Rep. Stat. Appl. Res., JUSE 11, 64–90.Google Scholar
Gumbel, E. J. (1958) Statistics of Extremes. Columbia University Press, New York.CrossRefGoogle Scholar
Gupta, S. S. (1961) Percentage points and modes of order statistics from the normal distribution. Ann. Math. Statist. 32, 888–893.CrossRefGoogle Scholar
Haldane, J. B. S. and Jayakar, S. D. (1963) The distribution of extremal and nearly extremal values in samples from a normal population. Biometrika 50, 89–94.Google Scholar
Hall, P. (1979) On the rate of convergence of normal extremes. J. Appl. Prob. 16, 433–439.Google Scholar
Pearson, E. S. and Hartley, H. O. (1972) Biometrika Tables for Statisticians, Vol. II. Cambridge University Press, London.Google Scholar
Tippett, L. H. C. (1925) On the extreme individuals and the range of samples taken from a normal population. Biometrika 17, 364–387.CrossRefGoogle Scholar