A Procedure for Estimating the Variance of the Population Mean in Rejective Sampling

Abstract

Rejective sampling was first introduced by Hájek in 1964 as a way to select a sample consisting uniquely of distinct units. If n denotes the fixed sample size, the n units are drawn independently with probabilities that may vary from unit to unit and the samples in which all units are not distinct are rejected. More generally, in rejective sampling, we select repeated samples according to a basic sampling design until a selected sample meets a specified balancing tolerance. Given a set of auxiliary variables, we consider a procedure in which the probability sample is rejected unless the sample mean of the auxiliary variables is within a specified distance of its corresponding population mean. The procedure represents an alternative to the well-known balanced cube method. In this article, we propose an estimator of the variance under the rejective sampling design. We also present the results of a Monte Carlo simulation study.

Keywords

Balanced sampling rejective sampling normality assumption

References

Chauvet, G., D. Haziza, and E. Lesage. 2017. “Examining Some Aspects of Balanced Sampling in Surveys.” Statistica Sinica 27: 313–334. DOI: http://dx.doi.org/10.5705/ss.2013.244.

Deville, J.-C. and C.E. Särndal. 1992. “Calibration estimators in survey sampling.” Journal of the American Statistical Association 87: 376–382. DOI: http://doi.org/10.2307/2290268.

Deville, J.-C. and Y. Tillé. 2004. “Efficient balanced sampling: The cube method.” Biometrika 91: 893–912. DOI: http://doi.org/10.1093/biomet/91.4.893.

Deville, J.-C. and Y. Tillé. 2005. “Variance approximation under balanced sampling.” Journal of statistical planning and inference 128: 569–591. DOI: https://doi.org/10.1016/j.jspi.2003.11.011.

Fuller, W.A. 2009. “Some design properties of a rejective sampling procedure.” Biometrika 96: 933–944. DOI: https://doi.org/10.1093/biomet/asp042.

Fuller, W.A., J.C. Legg, and Y. Li. 2017. “Bootstrap variance estimation for rejective sampling.” Journal of the American Statistical Association 112: 1562–1570. DOI: https://doi.org/10.1080/01621459.2016.1222285.

Hájek, J. 1964. “Asymptotic theory of rejective sampling with varying probabilities from a finite population.” Ann. Math. Statist. 35: 1491–1523. DOI: https://doi.org/10.1214/aoms/1177700375.

Hájek, J. 1981. Sampling from a finite population. Statistics: Textbooks and Monographs 37. New York: Marcel Dekker Inc.

Horvitz, D.G. and D.J. Thompson. 1952. “A generalization of sampling without replacement from a finite universe.” Journal of the American Statistical Association 47: 663–685. DOI: https://doi.org/10.2307/2280784.

10.

Huang, E.T. and W.A. Fuller. 1978. “Nonnegative regression estimation for sample survey data.” Proceedings of the Social Statistics Section, American Statistical Association. Alexandria, VA, 300–305. Available at: https://lib.dr.iastate.edu/rtd/6460 (accessed February 2020).

11.

Legg, J.C. and C.L. Yu. 2010. “A comparison of sample set restriction procedures.” Survey Methodology 36: 69–79. Available at: https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X201000111249 (accessed February 2020).

12.

Matei, A. and Y. Tillé. 2005. “Evaluation of variance approximations and estimators in maximum entropy sampling with unequal probability and fixed sample size.” Journal of Official Statistics 21(4): 543–570. Available at: https://www.scb.se/contentassets/ca21efb41fee47d293bbee5bf7be7fb3/evaluation-of-variance-approximations-and-estimators-in-maximum-entropy-sampling-with-unequal-probability-and-fixed-sample-size.pdf (accessed February 2020).

13.

Särndal, C.-E., B. Swenson, and J. Wretman. 1992. Model Assisted Survey Sampling. New York: Springer-Verlag.