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

Abstract

To each of the items $1,2,\cdots, N$ 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 $\Delta(s; n)$ be the probability that item $s$ is obtained in the first $n$ draws and let $Z_n$ 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 $\Delta(s; n), EZ_n$ and $\operatorname{Cov}(Z_{n_1}, Z_{n_2})$. Furthermore it is shown that, still under general conditions, the joint distribution of $Z_{n_1}, Z_{n_2},\cdots, Z_{n_d}$ is asymptotically normal. The general results are then applied to obtain asymptotic results for a "quasi"-Horvitz-Thompson estimator of the population total.