Electronic Journal of Statistics

Size constrained unequal probability sampling with a non-integer sum of inclusion probabilities

Abstract

More than 50 methods have been developed to draw unequal probability samples with fixed sample size. All these methods require the sum of the inclusion probabilities to be an integer number. There are cases, however, where the sum of desired inclusion probabilities is not an integer. Then, classical algorithms for drawing samples cannot be directly applied. We present two methods to overcome the problem of sample selection with unequal inclusion probabilities when their sum is not an integer and the sample size cannot be fixed. The first one consists in splitting the inclusion probability vector. The second method is based on extending the population with a phantom unit. For both methods the sample size is almost fixed, and equal to the integer part of the sum of the inclusion probabilities or this integer plus one.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 1477-1489.

Dates
First available in Project Euclid: 31 August 2012

https://projecteuclid.org/euclid.ejs/1346421601

Digital Object Identifier
doi:10.1214/12-EJS719

Mathematical Reviews number (MathSciNet)
MR2988455

Zentralblatt MATH identifier
1295.62010

Subjects
Primary: 62D05: Sampling theory, sample surveys

Citation

Grafström, Anton; Qualité, Lionel; Tillé, Yves; Matei, Alina. Size constrained unequal probability sampling with a non-integer sum of inclusion probabilities. Electron. J. Statist. 6 (2012), 1477--1489. doi:10.1214/12-EJS719. https://projecteuclid.org/euclid.ejs/1346421601

References

• Aires, N. (2000)., Techniques to calculate exact inclusion probabilities for conditional Poisson sampling and Pareto $\pi ps$ sampling designs. Doctoral thesis, Chalmers University of Technology and Göteborg University, Göteborg, Sweden.
• Antal, E. and Tillé, Y. (2011). A direct bootstrap method for complex sampling designs from a finite population., Journal of the American Statistical Association, 106(494), 534–543.
• Bondesson, L. and Grafström, A. (2011). An extension of Sampford’s method for unequal probability sampling., Scandinavian Journal of Statistics, 38(2), 377–392.
• Berger, Y. G. (1998). Rate of convergence to normal distribution for the Horvitz-Thompson estimator., Journal of Statistical Planning and Inference, 67, 209–226.
• Brewer, K. R. W. and Donadio, M. E. (2003). The high entropy variance of the Horvitz-Thompson estimator., Survey Methodology, 29, 189–196.
• Brewer, K. R. W. and Hanif, M. (1983)., Sampling with Unequal Probabilities. Springer, New York.
• Brown, L. D. (1986)., Fundamentals of Statistical Exponential Families: With Applications in Statistical Decision Theory. Hayward, CA: Institute of Mathematical Statistics.
• Chen, S. X., Dempster, A. P., and Liu, J. S. (1994). Weighted finite population sampling to maximize entropy., Biometrika, 81, 457–469.
• Darroch, J. N. and Ratcliff, D. (1972). Generalized iterative scaling for log-linear models., Annals of Mathematical Statistics, 43, 1470–1480.
• Deville, J.-C. (2000). Note sur l’algorithme de Chen, Dempster et Liu. Technical report, CREST-ENSAI, Rennes, France.
• Deville, J.-C. and Tillé, Y. (1998). Unequal probability sampling without replacement through a splitting method., Biometrika, 85, 89–101.
• Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population., Annals of Mathematical Statistics, 35, 1491–1523.
• Hájek, J. (1981)., Sampling from a Finite Population. New York: Marcel Dekker.
• Hardy, G. H., Littlewood, J. E. and Pólya, G. (1956)., Inequalities. Cambridge Univ. Press.
• Qualité, L. (2008). A comparison of conditional Poisson sampling versus unequal probability sampling with replacement., Journal of Statistical Planning and Inference, 138, 1428–1432.
• Sampford, M. R. (1967). On sampling without replacement with unequal probabilities of selection., Biometrika, 54, 499–513.
• Särndal, C. -E. and Swensson, B. and Wretman, J. H. (1992)., Model Assisted Survey Sampling. Springer Verlag, New York.
• Tillé, Y. (2006)., Sampling Algorithms. Springer, New York.