A Note on Sampling with Replacement

Abstract

Suppose a finite population is sampled with replacement until the sample contains a fixed number $n$ of distinct units. Let $v$ denote the total number of draws. It is known that $\bar{y}_n$, the mean for the $n$ distict units, and $\bar{y}_v$, the total sample mean, are both unbiased estimators of the population means and that $V(\bar{y}_n) \leqq V (\bar{y}_v)$. In this paper the relative difference $\delta = \lbrack V(\bar{y}_v) - V)\bar{y}_n)\rbrack/V(\bar{y}_n)$ is approximate by a quantity $\delta_1$ which is easy to compute. Upper and lower bounds for $\delta - \delta_1$ are given and it is shown that $\delta < (\lambda + \varepsilon_n) f$ for $n \geqq 3$ and $f \leqq \frac{3}{4}$, where $f = n/N, N$ is the population size, $\lambda = \lbrack (1 - f)^{-\frac{1}{2}} - 1 \rbrack/f,$ and $\varepsilon_n = (1 - f)^{-1}/(n - 1)$.