Berger, Y.G. (1998) Rate of convergence to normal distribution for the Horvitz-Thompson estimator. Journal of Statistical Planning and Inference, 67 (2), 209-226. (doi:10.1016/S0378-3758(97)00107-9).
Abstract
Sampling distinct units from a population with unequal probabilities without replacement is a problem often considered in the literature, e.g. Hanif and Brewer (1980). If we implement such a sampling design, we can estimate the total of an unknown characteristic by the Horvitz-Thompson estimator (1951). One of the aims of statistical inference in a sample survey is to have an asymptotic normal distribution for this estimator. Stenlund and Westlund (1975) examine this problem from an empirical point of view. In this paper, we give a theoretical framework where we show that this problem can be solved by maximizing entropy. Hajek (1981, p. 33) conjectured one of this fact but without a formal expression. Hajek (1964) gives a necessary and sufficient condition for the asymptotic normality of the Horvitz-Thompson estimator, if the rejective sampling is performed. In this work, we give a rate of convergence for any kind of sampling. We apply our results to the Rao (1965) and Sampford (1967) sampling and to the successive sampling (Hajek, 1964).
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