Tail approximations for samples from a finite population with applications to permutation tests
ESAIM: Probability and Statistics, Volume 16 (2012), pp. 425-435.

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

DOI: 10.1051/ps/2010027
Classification: 62E20,  60F05
Keywords: Cramér large deviation, saddlepoint approximations, moderate deviations, finite population, permutation tests 
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Hu, Zhishui; Robinson, John; Wang, Qiying. Tail approximations for samples from a finite population with applications to permutation tests. ESAIM: Probability and Statistics, Volume 16 (2012), pp. 425-435. doi : 10.1051/ps/2010027. http://archive.numdam.org/articles/10.1051/ps/2010027/

[1] G.J. Babu and Z.D. Bai, Mixtures of global and local Edgeworth expansions and their applications. J. Multivariate Anal. 59 (1996) 282-307. | MR | Zbl

[2] P.J. Bickel and W.R. Van Zwet, Asymptotic expansions for the power of distribution-free tests in the two-sample problem. Ann. Statist. 6 (1978) 937-1004. | MR | Zbl

[3] A. Bikelis, 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). | MR | Zbl

[4] M. Bloznelis, One and two-term Edgeworth expansion for finite population sample mean. Exact results I. Lith. Math. J. 40 (2000) 213-227. | MR | Zbl

[5] M. Bloznelis, One and two-term Edgeworth expansion for finite population sample mean. Exact results II. Lith. Math. J. 40 (2000) 329-340. | MR | Zbl

[6] J.G. Booth and R.W. Butler, Randomization distributions and saddlepoint approximations in generalized linear models. Biometrika 77 (1990) 787-796. | MR | Zbl

[7] P. Erdös and A. Rényi, On the central limit theorem for samples from a finite population. Publ. Math. Inst. Hungarian Acad. Sci. 4 (1959) 49-61. | MR | Zbl

[8] J. Hájek, Limiting distributions in simple random sampling for a finite population. Publ. Math. Inst. Hugar. Acad. Sci. 5 (1960) 361-374. | MR | Zbl

[9] T. Höglund, Sampling from a finite population. A remainder term estimate. Scand. J. Stat. 5 (1978) 69-71. | MR | Zbl

[10] Z. Hu, J. Robinson and Q. Wang, Crameŕ-type large deviations for samples from a finite population. Ann. Statist. 35 (2007) 673-696. | MR | Zbl

[11] J. Robinson, Large deviation probabilities for samples from a finite population. Ann. Probab. 5 (1977) 913-925. | MR | Zbl

[12] J. Robinson, An asymptotic expansion for samples from a finite population. Ann. Statist. 6 (1978) 1004-1011. | MR | Zbl

[13] J. Robinson, Saddlepoint approximations for permutation tests and confidence intervals. J. R. Statist. Soc. B 20 (1982) 91-101. | MR | Zbl

[14] J. Robinson, T. Hoglund, L. Holst and M.P. Quine, On approximating probabilities for small and large deviations in Rd. Ann. Probab. 18 (1990) 727-753. | MR | Zbl

[15] S. Wang, Saddlepoint expansions in finite population problems. Biometrika 80 (1993) 583-590. | MR | Zbl

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