In surveying finite populations, the simplest strategy to estimate a population total without bias is to employ Simple Random Sampling (SRS) with replacement (SRSWR) and the expansion estimator based on it. Anything other than that including SRS Without Replacement (SRSWOR) and usage of the expansion estimator is a complex strategy. We examine here (1) if from a complex sample at hand a gain in efficiency may be unbiasedly estimated comparing the ”rival population total-estimators” for the competing strategies and (2) how suitable model-expected variances of rival estimators compete in magnitude as examined numerically through simulations.
Des Raj and symmetrized Des Raj estimator and associated variance, Hansen- Hurwitz estimation and variance, Hartley-Ross, Horvitz-Thompson, Lahiri-Midzuno-Sen, Murthy, Rao-Hartley-Cochran procedures vis-a-vis SRSWOR and SRSWR
Bayless, D. L. and Rao, J. N. K., (1970). An empirical study of stabilities of estimators and variance estimators in unequal probability sampling (n=3or 4), Jour. Amer. Stat.Assoc. 65, pp. 1645–1667.
Chaudhuri, A., (2010). Essentials of survey sampling, Prentice Hall of India, New Delhi.
Chaudhuri, A. and Arnab, R., (1979). On the relative efficiencies of sampling strategies under a super-population model, Sankhya, Ser. C. 41, pp. 40–43.
Cochran, W., G., (1977). Sampling Techniques. John Wiley and Sons. New York.
Des Raj, (1956). Some estimators in sampling with varying probabilities without replacement, Jour. Amer. Stat. Assocn. 51, pp. 269–284.
Hansen, M. H. and Hurwitz, W. N., (1943). On the theory of sampling from finite populations, Ann. Math. Stat, 14, pp. 333–362.
Hartley, H. O. and Ross, A., (1954). Unbiased ratio estimators, Nature, 174, pp. 270–271.
Horvitz, D. G. and Thompson, D. J., (1952). A generalization of sampling without replacement from a finite universe, Jour. Amer. Stat. Assoc., 47, 663–685.
Lahiri, D. B., (1951). A method of sample selection providing unbiased ratio estimates, Bull. Int. Stat. Inst. 33(2), pp. 133–140.
Midzuno, H., (1952). An Outline of the theory of sampling systems, Annals. Inst. Stat. Math, 1, pp. 149–156.
Murthy, M. N., (1957). Ordered and unordered estimators in sampling without replacement, Sankhya, 18, pp. 379–390.
Rao, J. N. K., (1961). On the estimate of variance in unequal probability sampling, Annals. Inst. Stat. Math, 13, pp. 57–60.
Rao, J.N.K. and Bayless, D. L., (1969). An empirical study of the stabilities of estimators and variance estimators in unequal probability of two units per stratum, Jour. Amer. Stat. Assoc. 64, pp. 540–549.
Rao, J. N. K., Hartley, H. O., Cochran, N.G., (1962). On a simple procedure of unequal probability sampling without replacement, Jour. Roy. Stat. Soc. B. 24, pp. 482–491.
Rao, T. J., (1967). On the choice of a strategy for a ratio method of estimation, Jour. Roy.Stat.Soc. B. 29, pp. 392–397.
Roychoudhury, D. K., (1957). Unbiased sampling design using information provided by linear function of auxiliary variate, Chapter 5, thesis for Associateship of Indian Statistical Institute, Kolkata.
Sarndal. C. E., Swensson, B and Wretman, J., (1992). Model Assisted Survey Sampling, Springer Verlag, Heidelberg.
Sen, A. R., (1953). On the estimator of the variance in sampling with varying probabilities, J. Ind. Soc. Agri. Stat. 5(2), pp. 119–127.
Smith, H. F., (1938). An empirical law describing heterogeneity in the yields of agricultural crops, Jour. Agri. Sci. 28, pp. 1–23.