estimates.
RANSYS is commonly used due to its simplicity and the ease with which it is possible to rotate samples. Although the implementation of CPS is not as simple or as fast as RANSYS, it is still clearly feasible for moderately large populations. In addition, it is also possible to rotate samples under CPS. The choice of sampling algorithm may also depend upon the preferred approximate estimator. For instance, if the conservative estimator of Vˆ˜BR−Dev is chosen then RANSYS should be used. Overall there is no clear choice between RANSYS and CPS; both have their advantages.
5.3
Further Research
This thesis concludes by discussing further areas of research. There were a few interesting discoveries in Chapter 3 which were not examined further, but should definitely be considered. The first is to compare the accuracy of Hartley and Rao’s full approximation of the joint inclusion probabilities with the simpler third-order approximation, the latter approximation may in fact be more accurate. The second is to determine the number of simulations which are needed to ensure that the RBs are consistent over different sets of simulations for the same population. This would enable a firm conclusion to be arrived at about the behaviour of the variance estimators over all possible samples in a population. As simulations are the main approach to comparing the behaviour of variance estimators, it is important that a sufficient number of simulations are used to ensure the comparisons are reliable.
In regards to the relationship between the entropy and the variance of sampling designs, other designs should also be considered. In particular the relationship
5.3 Further Research 5 CONCLUSION
between the entropy and the variance should be analysed for individual samples, not just the sampling design as a whole. This may assist in finding sampling algorithms which guarantee a high entropy as well as a low variance.
REFERENCES REFERENCES
References
Aires, N.(1999). Algorithms to Find Exact Inclusion Probabilities for Conditional
Poisson Sampling and Paretoπps Sampling Designs, Methodology and Computing in Applied Probability, 4, 457–469.
Aires, N.(2000). Comparisons between Conditional Poisson Sampling and Pareto
πps Sampling Designs, Journal of Statistical Planning and Inference, 82, 1–15.
Asok, C. and Sukhatme, B. (1976). On Sampford’s Procedure of Unequal
Probability Sampling Without Replacement, Journal of the American Statistical Association, 71, 912–918.
Bellhouse, D. and Rao, J. (1975). Systematic Sampling in the Presence of a
Trend, Biometrika, 62, 690–697.
Berger (2004). A Simple Variance Estimator for Unequal Probability Sampling
Without Replacement, Journal of Applied Statistics, 31, 305–315.
Brewer, K. (2002). Combined Survey Sampling Inference; Weighing Basu’s
Elephants, London: Arnold.
Brewer, K. and Donadio, M. E. (2003). The High Entropy Variance of the
Horvitz-Thompson Estimator, Survey Methodology, 29, 189–196.
Brewer, K. and Hanif, M. (1983). Sampling with Unequal Probabilities, New
York: Springer-Verlag.
Chao, M. (1982). A General Purpose Unequal Probability Sampling Plan,
Biometrika, 69, 653–656.
Chen, X.-H., Dempster, A. P., and Liu, J. S. (1994). Weighted Finite
REFERENCES REFERENCES
Deville, J.-C. (1999). Estimation de la Variance pour les Enquˆetes en Deux
Phases, note Interne Manusctrite. France: INSEE.
Deville, J.-C.(2000). Note sur l’Algorithme de Chen, Technical report, Dempster
et Liu, France CREST-ENSAI.
Donadio, M. E. (2002). Variance Estimation in πps Sampling, Master’s thesis,
The University of Melbourne.
Dupaˇcov´a, J. (1979). A note on Rejective Sampling, Contributions to Statistics,
Jaroslav H´ajek Memorial Volume, Prague: Reidal, Holland and Academia.
Foreman, E.(1991).Survey Sampling Principles, New York: Marcel Dekker, Inc.
Goodman, R. and Kish, L. (1950). Controlled Selection - a Technique in
Probability Sampling, Journal of the American Statistical Association, 45, 350– 372.
H´ajek, J. (1964). Asymptotic Theory of Rejection Sampling with Varying
Probabilities from a Finite Population, Annals of Mathematical Statistics, 35, 1491 – 1523.
H´ajek, J. (1981). Sampling from a Finite Population, New York: Marcel Dekker,
Inc.
Hansen, M. and Hurwitz, W. (1943). On the Theory of Sampling from Finite
Populations, Annals of Mathematical Statistics,14, 333–362.
Hartley, H. and Rao, J. (1962). Sampling with Unequal Probabilities and
without Replacement,Annals of Mathematical Statistics, 33, 350–374.
Horvitz, D. and Thompson, D. (1952). A Generalisation of Sampling Without
Replacement from a Finite Universe, Journal of the American Statistical Association, 47, 663–685.
REFERENCES REFERENCES
Matei, A. and Till´e, Y. (2005). Evaluation of Variance Approximations and
Estimators in Maximum Entropy Sampling with Unequal Probability and Fixed Sample Size, Journal of Official Statistics, 21, 543–570.
Ohlsson, E.(1995). Coordination of Samples Using Permanent Random Numbers,
in Business Survey Methods, New York, Wiley.
Ros´en, B. (1991). Variance Estimation for Systematic pps-sampling, Technical
Report 15, Statistics Sweden.
S¨arndal, C.-E., Swensson, B., and Wretman, J. (2003). Model Assisted
Survey Sampling, New York: Springer.
Sen, A. (1953). On the Estimate of the Variance in Sampling with Varying
Probabilities,Journal of the Indian Society of Agricultural Statistics,5, 119–127.
Shannon, C. E. (1948). A Mathematical Theory of Communication, The Bell
System, 27, 379–423,623–656.
Till´e, Y. (1996a). An Elimination Procedure for Unequal Probability Sampling
Without Replacement, Biometrika,83, 238–241.
Till´e, Y. (1996b). Some Remarks on Unequal Probability Sampling Designs
Without Replacement, Annales D’ ´Economie et de Statistique, 44, 177–189.
Till´e, Y. (2006). Sampling Algorithms, New York: Springer.
Till´e, Y. and Matei, A. (2006). Sampling: Survey Sampling, Department of
Statistics and Mathematics of the WU Wien, Retrieved September 2, 2006, http://cran.r-project.org.
Wikipedia(2006).Information Entropy, Wikipedia, Retrieved September 8, 2006,
REFERENCES REFERENCES
Yates, F. and Grundy, P.(1953). Selection Without Replacement from Within
Strata with Probabilities Proportion to Size, Journal of the Royal Statistical Society, 15, 235–261, series B.
A SIMULATION CODE