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Asymptotics of multivariate contingency tables with fixed marginals. (English) Zbl 1432.62170

Summary: We consider the asymptotic distribution of a cell in a \(2 \times \cdots \times 2\) contingency table as the fixed marginal totals tend to infinity. The asymptotic order of the cell variance is derived and a useful diagnostic is given for determining whether the cell has a Poisson limit or a Gaussian limit. There are three forms of Poisson convergence. The exact form is shown to be determined by the growth rates of the two smallest marginal totals. The results are generalized to contingency tables with arbitrary sizes and are further complemented with concrete examples.

MSC:

62H17 Contingency tables
60F05 Central limit and other weak theorems
60C05 Combinatorial probability
62E20 Asymptotic distribution theory in statistics
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[1] Arratia, Richard; Goldstein, Larry; Gordon, Louis, Two moments suffice for Poisson approximations: the Chen-Stein method, Ann. Probab., 9-25, (1989) · Zbl 0675.60017
[2] Barbour, A. D.; Holst, Lars, Some applications of the Stein-Chen method for proving Poisson convergence, Adv. Appl. Probab., 21, 1, 74-90, (1989) · Zbl 0673.60023
[3] Barbour, Andrew D.; Holst, Lars; Janson, Svante, Poisson approximation, (1992), Clarendon Press Oxford · Zbl 0746.60002
[4] Cekanavicius, Vydas; Kruopis, Julius, Signed Poisson approximation: a possible alternative to normal and Poisson laws, Bernoulli, 6, 4, 591-606, (2000) · Zbl 0976.60035
[5] Daly, Fraser, Johnson, Oliver, 2017. Relaxation of monotone coupling conditions: Poisson approximation and beyond, arXiv preprint arXiv:1706.04064; Daly, Fraser, Johnson, Oliver, 2017. Relaxation of monotone coupling conditions: Poisson approximation and beyond, arXiv preprint arXiv:1706.04064 · Zbl 1402.62022
[6] Daly, Fraser; Lefèvre, Claude; Utev, Sergey, Stein’s method and stochastic orderings, Adv. Appl. Probab., 44, 2, 343-372, (2012) · Zbl 1276.62015
[7] Darroch, John N., The multiple-recapture census: I. estimation of a closed population, Biometrika, 45, 3/4, 343-359, (1958) · Zbl 0099.14703
[8] DasGupta, Anirban, The matching, birthday and the strong birthday problem: a contemporary review, J. Statist. Plann. Inference, 130, 1, 377-389, (2005) · Zbl 1089.60502
[9] Diaconis, Persi; Holmes, Susan, A Bayesian peek into Feller volume I, Sankhyā, 820-841, (2002) · Zbl 1192.60003
[10] Erhardsson, Torkel, Steins method for Poisson and compound Poisson, (An Introduction To Stein’S Method, Vol. 4, (2005)), 61
[11] Harris, Bernard, Poisson limits for generalized random allocation problems, Statist. Probab. Lett., 8, 2, 123-127, (1989) · Zbl 0679.60016
[12] Holst, Lars, On matrix occupancy, committee, and capture-recapture problems, Scand. J. Stat., 139-146, (1980) · Zbl 0445.62024
[13] Holst, Lars, On birthday, collectors’, occupancy and other classical urn problems, Int. Statist. Rev./Rev. Int. Statist., 15-27, (1986) · Zbl 0594.60014
[14] Joag-Dev, Kumar; Proschan, Frank, Negative association of random variables with applications, Ann. Statist., 286-295, (1983) · Zbl 0508.62041
[15] Kolchin, Valentin Fedorovich, Sevastyanov, Boris Aleksandrovich, Chistyakov, Vladimir Pavlovich, 1978. Random allocations.; Kolchin, Valentin Fedorovich, Sevastyanov, Boris Aleksandrovich, Chistyakov, Vladimir Pavlovich, 1978. Random allocations. · Zbl 0376.60003
[16] Kou, S. G.; Ying, Z., Asymptotics for a 2\(\times\) 2 table with fixed margins, Statist. Sinica, 809-829, (1996) · Zbl 1076.62510
[17] Lareida, Andri; Hoßfeld, Tobias; Stiller, Burkhard, The BitTorrent peer collector problem, (2017 IFIP/IEEE Symposium on Integrated Network and Service Management, (IM), (2017), IEEE), 449-455
[18] Mitwalli, Saleh M., An occupancy problem with group drawings of different sizes, Math. Slovaca, 52, 2, 235-242, (2002) · Zbl 1008.60018
[19] Smythe, R. T., Generalized coupon collection: the superlinear case, J. Appl. Probab., 48, 1, 189-199, (2011) · Zbl 1213.60050
[20] Stadje, Wolfgang, The collector’s problem with group drawings, Adv. Appl. Probab., 22, 4, 866-882, (1990) · Zbl 0711.60013
[21] Vatutin, V. A.; Mikhailov, V. G., Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles, Theory Probab. Appl., 27, 4, 734-743, (1983) · Zbl 0536.60017
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