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Asymptotic Theory for Successive Sampling with Varying Probabilities Without Replacement, I

Bengt Rosen
The Annals of Mathematical Statistics
Vol. 43, No. 2 (Apr., 1972), pp. 373-397
Stable URL: http://www.jstor.org/stable/2239977
Page Count: 25
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Asymptotic Theory for Successive Sampling with Varying Probabilities Without Replacement, I
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Abstract

To each of the items 1,2,⋯, N in a finite population there is associated a variate value. The population is sampled by successive drawings without replacement in the following way. At each draw the probability of drawing item s is proportional to a number $p_s > 0$ if item s remains in the population and is 0 otherwise. Let Δ(s; n) be the probability that item s is obtained in the first n draws and let Zn be the sum of the variate values obtained in the first n draws. Asymptotic formulas, valid under general conditions when n and N both are "large", are derived for Δ(s; n), EZn and $\operatorname{Cov}(Z_{n_1}, Z_{n_2})$. Furthermore it is shown that, still under general conditions, the joint distribution of Zn1 , Zn2 ,⋯, Znd is asymptotically normal. The general results are then applied to obtain asymptotic results for a "quasi"-Horvitz-Thompson estimator of the population total.

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