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March, 1975 A Note on Sampling with Replacement
E. Benton Cobb
Ann. Statist. 3(2): 500-503 (March, 1975). DOI: 10.1214/aos/1176343079


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 $\varepsilon_n = (1 - f)^{-1}/(n - 1)$.


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E. Benton Cobb. "A Note on Sampling with Replacement." Ann. Statist. 3 (2) 500 - 503, March, 1975.


Published: March, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0328.62007
MathSciNet: MR378167
Digital Object Identifier: 10.1214/aos/1176343079

Primary: 62D05
Secondary: 62F10

Keywords: estimation of the mean of a finite population , Sampling with replacement until the sample contains $n$ units

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 2 • March, 1975
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