The Annals of Statistics

Cramér-type large deviations for samples from a finite population

Zhishui Hu, John Robinson, and Qiying Wang

Full-text: Open access

Abstract

Cramér-type large deviations for means of samples from a finite population are established under weak conditions. The results are comparable to results for the so-called self-normalized large deviation for independent random variables. Cramér-type large deviations for the finite population Student t-statistic are also investigated.

Article information

Source
Ann. Statist. Volume 35, Number 2 (2007), 673-696.

Dates
First available in Project Euclid: 5 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1183667288

Digital Object Identifier
doi:10.1214/009053606000001343

Mathematical Reviews number (MathSciNet)
MR2336863

Zentralblatt MATH identifier
1272.68116

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

Keywords
Cramér large deviation moderate deviation finite population

Citation

Hu, Zhishui; Robinson, John; Wang, Qiying. Cramér-type large deviations for samples from a finite population. Ann. Statist. 35 (2007), no. 2, 673--696. doi:10.1214/009053606000001343. https://projecteuclid.org/euclid.aos/1183667288


Export citation

References

  • Babu, G. J. and Bai, Z. D. (1996). Mixtures of global and local Edgeworth expansions and their applications. J. Multivariate Anal. 59 282--307.
  • Babu, G. J. and Singh, K. (1985). Edgeworth expansions for sampling without replacement from finite populations. J. Multivariate Anal. 17 261--278.
  • Bickel, P. J. and van Zwet, W. R. (1978). Asymptotic expansions for the power of distribution-free tests in the two-sample problem. Ann. Statist. 6 937--1004.
  • Bikelis, A. (1969). On the estimation of the remainder term in the central limit theorem for samples from finite populations. Studia Sci. Math. Hungar. 4 345--354. (In Russian.)
  • Bloznelis, M. (1999). A Berry--Esseen bound for finite population Student's statistic. Ann. Probab. 27 2089--2108.
  • Bloznelis, M. (2000). One and two-term Edgeworth expansions for finite population sample mean. Exact results. I. Lithuanian Math. J. 40 213--227.
  • Bloznelis, M. (2000). One- and two-term Edgeworth expansions for finite population sample mean. Exact results. II. Lithuanian Math. J. 40 329--340.
  • Bloznelis, M. (2003). An Edgeworth expansion for Studentized finite population statistics. Acta Appl. Math. 78 51--60.
  • Bloznelis, M. and Götze, F. (2000). An Edgeworth expansion for finite-population $U$-statistics. Bernoulli 6 729--760.
  • Bloznelis, M. and Götze, F. (2001). Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Statist. 29 899--917.
  • Dai, W. and Robinson, J. (2001). Empirical saddlepoint approximations of the Studentized mean under simple random sampling. Statist. Probab. Lett. 53 331--337.
  • de La Pena, V. H., Klass, M. J. and Lai, T. L. (2004). Self-normalized processes: Exponential inequalities, moment bounds and iterated logarithm laws. Ann. Probab. 32 1902--1933.
  • Efron, B. (1969). Student's $t$-test under symmetry conditions. J. Amer. Statist. Assoc. 64 1278--1302.
  • Erdös, P. and Rényi, A. (1959). On the central limit theorem for samples from a finite population. Publ. Math. Inst. Hungarian Acad. Sci. 4 49--61.
  • Hájek, J. (1960). Limiting distributions in simple random sampling for a finite population. Publ. Math. Inst. Hungarian Acad. Sci. 5 361--374.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13--30.
  • Höglund, T. (1978). Sampling from a finite population: A remainder term estimate. Scand. J. Statist. 5 69--71.
  • Hu, Z., Robinson, J. and Wang, Q. (2006). Cramér-type large deviations for samples from a finite population. Research Report 2, Univ. Sydney. Available at www.maths.usyd.edu.au/u/pubs/publist/preprints/2006/hu-2.html.
  • Jing, B.-Y., Shao, Q.-M. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167--2215.
  • Kokic, P. N. and Weber, N. C. (1990). An Edgeworth expansion for $U$-statistics based on samples from finite populations. Ann. Probab. 18 390--404.
  • Nandi, H. K. and Sen, P. K. (1963). On the properties of $U$-statistics when the observations are not independent. II. Unbiased estimation of the parameters of a finite population. Calcutta Statist. Assoc. Bull. 12 124--148.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.
  • Rao, C. R. and Zhao, L. C. (1994). Berry--Esseen bounds for finite-population $t$-statistics. Statist. Probab. Lett. 21 409--416.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • Robinson, J. (1977). Large deviation probabilities for samples from a finite population. Ann. Probab. 5 913--925.
  • Robinson, J. (1978). An asymptotic expansion for samples from a finite population. Ann. Statist. 6 1005--1011.
  • Zhao, L. C. and Chen, X. R. (1987). Berry--Esseen bounds for finite-population $U$-statistics. Sci. Sinica Ser. A 30 113--127.
  • Zhao, L. C. and Chen, X. R. (1990). Normal approximation for finite-population $U$-statistics. Acta Math. Appl. Sinica (English Ser.) 6 263--272.