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Tail approximations for samples from a finite population withapplications to permutation tests

Published online by Cambridge University Press:  31 August 2012

Zhishui Hu ,
John Robinson  and
Qiying Wang
Zhishui Hu
Affiliation:
Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, Anhui, P.R. China. [email protected]
John Robinson
Affiliation:
School of Mathematics and Statistics The University of Sydney, NSW, 2006, Australia; [email protected]; [email protected]
Qiying Wang
Affiliation:
School of Mathematics and Statistics The University of Sydney, NSW, 2006, Australia; [email protected]; [email protected]
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Abstract

This paper derives an explicit approximation for the tail probability of a sum of samplevalues taken without replacement from an unrestricted finite population. The approximationis shown to hold under no conditions in a wide range with relative error given in terms ofthe standardized absolute third moment of the population, β3N. This approximation is used to obtaina result comparable to the well-known Cramér large deviation result in the independentcase, but with no restrictions on the sampled population and an error term depending onlyon β3N. Application to permutation tests isinvestigated giving a new limit result for the tail conditional probability of thestatistic given order statistics under mild conditions. Some numerical results are givento illustrate the accuracy of the approximation by comparing our results to saddlepointapproximations requiring strong conditions.

Keywords

Cramér large deviationsaddlepoint approximationsmoderate deviationsfinite populationpermutation tests
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 425 - 435
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Babu, G.J. and Bai, Z.D., Mixtures of global and local Edgeworth expansions and their applications. J. Multivariate Anal. 59 (1996) 282–307. Google Scholar
Bickel, P.J. and van Zwet, W.R., Asymptotic expansions for the power of distribution-free tests in the two-sample problem. Ann. Statist. 6 (1978) 937–1004. Google Scholar
Bikelis, A., On the estimation of the remainder term in the central limit theorem for samples from finite populations. Stud. Sci. Math. Hung. 4 (1969) 345–354 (in Russian). Google Scholar
Bloznelis, M., One and two-term Edgeworth expansion for finite population sample mean. Exact results I. Lith. Math. J. 40 (2000) 213–227. Google Scholar
Bloznelis, M., One and two-term Edgeworth expansion for finite population sample mean. Exact results II. Lith. Math. J. 40 (2000) 329–340. Google Scholar
Booth, J.G. and Butler, R.W., Randomization distributions and saddlepoint approximations in generalized linear models. Biometrika 77 (1990) 787–796. Google Scholar
Erdös, P. and Rényi, A., On the central limit theorem for samples from a finite population. Publ. Math. Inst. Hungarian Acad. Sci. 4 (1959) 49–61. Google Scholar
Hájek, J., Limiting distributions in simple random sampling for a finite population. Publ. Math. Inst. Hugar. Acad. Sci. 5 (1960) 361–374. Google Scholar
Höglund, T., Sampling from a finite population. A remainder term estimate. Scand. J. Stat. 5 (1978) 69–71. Google Scholar
Hu, Z., Robinson, J. and Wang, Q., Crameŕ-type large deviations for samples from a finite population. Ann. Statist. 35 (2007) 673–696. Google Scholar
Robinson, J., Large deviation probabilities for samples from a finite population. Ann. Probab. 5 (1977) 913–925. Google Scholar
Robinson, J., An asymptotic expansion for samples from a finite population. Ann. Statist. 6 (1978) 1004–1011. Google Scholar
Robinson, J., Saddlepoint approximations for permutation tests and confidence intervals. J. R. Statist. Soc. B 20 (1982) 91–101. Google Scholar
Robinson, J., Hoglund, T., Holst, L. and Quine, M.P., On approximating probabilities for small and large deviations in R d. Ann. Probab. 18 (1990) 727–753. Google Scholar
Wang, S., Saddlepoint expansions in finite population problems. Biometrika 80 (1993) 583–590. Google Scholar