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The penultimate form of approximation to normal extremes

Published online by Cambridge University Press:  01 July 2016

Jonathan P. Cohen
Jonathan P. Cohen*
Affiliation:
Imperial College, London
*
∗Postal Address: Department of Mathematics (Statistics Section), Imperial College, Huxley Building, Queen's Gate, South Kensington, London SW7 2AZ, U.K.
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Abstract

Let Yn denote the largest of n independent N(0,1) random variables. It is shown that the error in approximating the distribution of Yn by the type III extreme value distribution exp {– (–Ax + B)k}, k > 0, is uniformly of order (log n)–2 if and only if the constants A, B and k satisfy certain conditions. In particular, this holds for the penultimate form of Fisher and Tippett (1928). Furthermore, two sufficient conditions are given so that these results can be extended to a stationary Gaussian sequence.

Keywords

NORMAL DISTRIBUTIONEXTREME VALUE DISTRIBUTIONRATE OF CONVERGENCESTATIONARY GAUSSIAN SEQUENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 324 - 339
Copyright
Copyright © Applied Probability Trust 1982 

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References

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