## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 2 (1975), 500-503.

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

#### Article information

**Source**

Ann. Statist., Volume 3, Number 2 (1975), 500-503.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343079

**Digital Object Identifier**

doi:10.1214/aos/1176343079

**Mathematical Reviews number (MathSciNet)**

MR378167

**Zentralblatt MATH identifier**

0328.62007

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62D05: Sampling theory, sample surveys

Secondary: 62F10: Point estimation

**Keywords**

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

#### Citation

Cobb, E. Benton. A Note on Sampling with Replacement. Ann. Statist. 3 (1975), no. 2, 500--503. doi:10.1214/aos/1176343079. https://projecteuclid.org/euclid.aos/1176343079