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Approximations to extreme tail probabilities for sampling without replacement

Published online by Cambridge University Press:  24 October 2008

M. Stone
M. Stone
Affiliation:
University College London
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Extract

Introduction and summary: In the distribution of the sum of n random variables

for which we suppose , the tail probability will be called an extreme tail probability if the normal approximation

is poor, when is asymptotically normal and n is large. This paper concerns the specialization of (1.1) in which

with a random permutation of nλ ones and n − nλ zeros, that is, in which T is the total for a simple random sample from the population with sampling fraction λ. In this case, we will always write T as Tλ. For brevity, we may write {c1,…, cn} for .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 66 , Issue 3 , November 1969 , pp. 587 - 606
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Cramér, H.Random variables and probability distributions. Cambridge Tract No. 36 (Cambridge University Press, 1937).Google Scholar
(2)Cramér, H.Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques. 736 (1938), 5–23.Google Scholar
(3)Dixon, W. J. and Massey, F. J.Introduction to statistical analysis (McGraw Hill; New York, 1957).CrossRefGoogle Scholar
(4)Feller, W.Generalization of a probability limit theorem of Cramér. Trans. Amer. Math. Soc. 54 (1943), 361–372.Google Scholar
(5)Hodges, J. L. and Lehmann, E. L.Basic concepts of probability and statistics (Holden-Day; San Francisco, 1964).Google Scholar
(6)Stone, M.Extreme tail probabilities for the null distribution of the two-sample Wilcoxon statistic. Biometrika 54 (1967), 629–640.CrossRefGoogle Scholar
(7)Stone, M.Extreme tail probabilities for sampling without replacement and exact Bahadur efficiency of the two-sample normal scores test. Biometrika 55 (1968), 315–319.CrossRefGoogle Scholar