Exact linear unbiased estimation in survey sampling. (With discussion). (English) Zbl 0989.62009

Sugden, R. A.; Smith, T. M. F.

J. Stat. Plann. Inference 102, No. 1, 25-45 (2002).

Summary: The concept of the linearity of estimators in finite population inference is not well defined. We propose that linearity should be defined for both estimators and estimands and that strict linearity should be related constructively to the property of unbiasedness. We establish the conditions for the class of general linear estimators, introduced by V.P. Godambe [J. R. Stat. Soc., Ser. B 17, 269-278 (1955; Zbl 0067.11406)], to be strictly linear. The ideas are extended to quadratic estimators of quadratic estimands. The implications are explored for ratio estimators, and domains of study, the latter extendible to poststratification.

MSC:

62D05

Sampling theory, sample surveys

References:

[1]	Basu, D., 1971. An essay on the logical foundations of survey sampling, part one. In: Godambe, V.P., Sprott, D.A. (Eds.), Foundations of Statistical Inference. Toronto, Holt, Rinehart and Winston, 203-242.
[2]	Cochran, W.G., Sampling techniques., (1977), Wiley New York · Zbl 0051.10707
[3]	Deville, J.C.; Särndal, C.E., Calibration estimators in survey sampling, J. amer. statist. assoc., 87, 376-382, (1992) · Zbl 0760.62010
[4]	Godambe, V.P., A unified theory of sampling from finite populations, J. roy. statist. soc. B., 17, 269-278, (1955) · Zbl 0067.11406
[5]	Haslett, S., The linear non-homogeneous estimator in sample surveys, Sankhyā: Indian J. of statistics, series B, 47, 101-117, (1985) · Zbl 0589.62008
[6]	Horvitz, D.G.; Thompson, D.J., A generalization of sampling without replacement from a finite universe, J. amer. statist. assoc., 47, 663-665, (1952) · Zbl 0047.38301
[7]	Rao, J.N.K., On deriving Mean square errors and their non-negative unbiased estimators in finite population sampling, J. Indian statist. assoc., 17, 125-136, (1979)
[8]	Rao, J.N.K., Estimating totals and distribution functions using auxiliary information at the estimation stage, J. official statist., 10, 153-165, (1994)
[9]	Smith, T.M.F., Sample surveys 1975-1990; an age of reconciliation?, Int. statist. rev., 62, 3-34, (1994) · Zbl 0925.62044
[10]	Sukhatme, P.V., Sampling theory of surveys with applications., (1954), Indian Society of Agricultural Statistics New Delhi, India · Zbl 0057.35703
[11]	Sukhatme, P.V.; Sukhatme, B.V., Sampling theory of surveys with applications, (1970), Asia Publishing House London · Zbl 0239.62008