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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.
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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 X 2 nn , 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

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 491 - 500
Copyright
Copyright © Applied Probability Trust 1980 

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