Tail approximations for samples from a finite population with applications to permutation tests
ESAIM: Probability and Statistics, Tome 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 : https://doi.org/10.1051/ps/2010027
Classification : 62E20,  60F05
Mots clés : Cramér large deviation, saddlepoint approximations, moderate deviations, finite population, permutation tests
@article{PS_2012__16__425_0,
     author = {Hu, Zhishui and Robinson, John and Wang, Qiying},
     title = {Tail approximations for samples from a finite population with applications to permutation tests},
     journal = {ESAIM: Probability and Statistics},
     pages = {425--435},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2010027},
     mrnumber = {2972501},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2010027/}
}
TY  - JOUR
AU  - Hu, Zhishui
AU  - Robinson, John
AU  - Wang, Qiying
TI  - Tail approximations for samples from a finite population with applications to permutation tests
JO  - ESAIM: Probability and Statistics
PY  - 2012
DA  - 2012///
SP  - 425
EP  - 435
VL  - 16
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2010027/
UR  - https://www.ams.org/mathscinet-getitem?mr=2972501
UR  - https://doi.org/10.1051/ps/2010027
DO  - 10.1051/ps/2010027
LA  - en
ID  - PS_2012__16__425_0
ER  - 
Hu, Zhishui; Robinson, John; Wang, Qiying. Tail approximations for samples from a finite population with applications to permutation tests. ESAIM: Probability and Statistics, Tome 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 1423736 | Zbl 0864.62006

[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 499567 | Zbl 0378.62047

[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 254902 | Zbl 0139.35302

[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 1803645 | Zbl 0966.62008

[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 1819377 | Zbl 0999.62008

[6] J.G. Booth and R.W. Butler, Randomization distributions and saddlepoint approximations in generalized linear models. Biometrika 77 (1990) 787-796. | MR 1086689 | Zbl 0709.62060

[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 107294 | Zbl 0086.34001

[8] J. Hájek, Limiting distributions in simple random sampling for a finite population. Publ. Math. Inst. Hugar. Acad. Sci. 5 (1960) 361-374. | MR 125612 | Zbl 0102.15001

[9] T. Höglund, Sampling from a finite population. A remainder term estimate. Scand. J. Stat. 5 (1978) 69-71. | MR 471130 | Zbl 0382.60028

[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 2336863 | Zbl 1117.62006

[11] J. Robinson, Large deviation probabilities for samples from a finite population. Ann. Probab. 5 (1977) 913-925. | MR 448498 | Zbl 0372.60037

[12] J. Robinson, An asymptotic expansion for samples from a finite population. Ann. Statist. 6 (1978) 1004-1011. | MR 499568 | Zbl 0387.60030

[13] J. Robinson, Saddlepoint approximations for permutation tests and confidence intervals. J. R. Statist. Soc. B 20 (1982) 91-101. | MR 655378 | Zbl 0487.62016

[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 1055431 | Zbl 0704.60018

[15] S. Wang, Saddlepoint expansions in finite population problems. Biometrika 80 (1993) 583-590. | MR 1248023 | Zbl 0785.62012

Cité par Sources :