A Berry-Esseen theorem for hypergeometric probabilities under minimal conditions

Lahiri, S.; Chatterjee, A.

doi:10.1090/S0002-9939-07-08676-5

A Berry-Esseen theorem for hypergeometric probabilities under minimal conditions
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by S. N. Lahiri and A. Chatterjee PDF

Proc. Amer. Math. Soc. 135 (2007), 1535-1545 Request permission

Abstract:

In this paper, we consider simple random sampling without replacement from a dichotomous finite population and derive a necessary and sufficient condition on the finite population parameters for a valid large sample Normal approximation to Hypergeometric probabilities. We then obtain lower and upper bounds on the difference between the Normal and the Hypergeometric distributions solely under this necessary and sufficient condition.

References

G. Jogesh Babu and Kesar Singh, Edgeworth expansions for sampling without replacement from finite populations, J. Multivariate Anal. 17 (1985), no. 3, 261–278. MR 813236, DOI 10.1016/0047-259X(85)90084-3
M. Bloznelis, A Berry-Esseen bound for finite population Student’s statistic, Ann. Probab. 27 (1999), no. 4, 2089–2108. MR 1742903, DOI 10.1214/aop/1022677563
Mindaugas Bloznelis and Friedrich Götze, An Edgeworth expansion for finite-population $U$-statistics, Bernoulli 6 (2000), no. 4, 729–760. MR 1777694, DOI 10.2307/3318517
Colin R. Blyth and Robert G. Staudte, Hypothesis estimates and acceptability profiles for $2\times 2$ contingency tables, J. Amer. Statist. Assoc. 92 (1997), no. 438, 694–699. MR 1467859, DOI 10.2307/2965717
Burstein, H. (1975). Finite population correction for binomial confidence limits. Journal of the American Statistical Association 70 67-69.
Paul Erdős and Alfréd Rényi, On the central limit theorem for samples from a finite population, Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 (1959), 49–61 (English, with Russian and Hungarian summaries). MR 107294
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
Jaroslav Hájek, Limiting distributions in simple random sampling from a finite population, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 361–374 (English, with Russian summary). MR 125612
Lahiri, S. N., Chatterjee, A. and Maiti, T. (2004). A Sub-Gaussian Berry-Esseen Theorem for the Hypergeometric probabilities. Preprint # 2004-21 (posted at http://seabiscuit.stat.iastate.edu/departmental/preprint/preprint.html#2004), Iowa State University, IA (also, posted as math.PR/0602276 at http://arxiv.org).
W. L. Nicholson, On the normal approximation to the hypergeometric distribution, Ann. Math. Statist. 27 (1956), 471–483. MR 87246, DOI 10.1214/aoms/1177728270
William G. Madow, On the limiting distributions of estimates based on samples from finite universes, Ann. Math. Statistics 19 (1948), 535–545. MR 29136, DOI 10.1214/aoms/1177730149
Patel, J. K. and Samaranayake, V. A. (1991). Prediction intervals for some discrete distributions. Journal of Quality Technology 23 270-278.
Seber, G. A. F. (1970). The effects of trap response on tag recapture estimates. Biometrics 26 13-22.
Sohn, S. Y. (1997). Accelerated life-tests for intermittent destructive inspection, with logistic failure-distribution. IEEE Transactions on Reliability 46 122-1295.
John P. Wendell and Josef Schmee, Exact inference for proportions from a stratified finite population, J. Amer. Statist. Assoc. 91 (1996), no. 434, 825–830. MR 1395749, DOI 10.2307/2291677
Wittes, J. T. (1972). On the bias and estimated variance of Chapman’s two-sample capture-recapture population estimate. Biometrics 28 592-597.