## The Annals of Mathematical Statistics

### Rotation Sampling

Albert Ross Eckler

#### Abstract

This paper shows how to find minimum-variance estimates of the mean $\alpha(t_i)$ of a time-dependent population, assuming that one is restricted to the class of linear unbiased estimates. Each minimum-variance estimate is based on a specified sample pattern (a set of sample values drawn from the population at one or more distinct times). Let the random variable $X_{ij}$ denote the value of element $j$ of the population at time $t_i$. The correlation between $X_{ik}$ and $X_{jk}$ is assumed to be $\rho^{|i-j|}$; the correlation between $X_{ij}$ and $X_{ik}$ is assumed to be zero; the variance of $X_{ij}$ is assumed to be $\sigma^2$ independent of time. Iterative methods are developed; the estimate of the population mean $\alpha(t_{i-1})$ is used in determining the population mean $\alpha(t_i)$. The paper discusses two important methods of sampling: in one-level rotation sampling, the statistician can add to the sample pattern only sample values that have been drawn from the population at the current time; in two-level (and higher-level) rotation sampling, the statistician can add earlier sample values as well as current ones to the pattern. Schematic sample patterns associated with these two methods are illustrated in (3.1) and (4.1). The optimum structure of a sample pattern is considered from two viewpoints: the variance of a pattern consisting of $n$ sample values drawn at each time $t_i$ is minimized; the number of sample values drawn at time $t_i$, is minimized while the variance of the minimum-variance estimate is held constant. Finally, the estimation problem is generalized to include minimum-variance estimates of linear functions of two or more population means at different times. In order to maintain continuity, this paper presents published results along with new results; the latter are summarized below. The paper clarifies Pat terson's fundamental method for finding minimum-variance linear unbiased estimates (Sections 2, 3) and extends his methods to two-level and three-level rotation sampling (Sections 4, 6, 8, 10, 13). The paper compares three methods of rotation sampling on a cost basis (Section 11) and shows how the one-level rotation sampling estimate of greatest practical interest can be derived from the two-level estimate (Sections 5, 14). Finally, the paper extends Cochran's work in determining optimum patterns for the one-level rotation sampling estimate (Section 9).

#### Article information

Source
Ann. Math. Statist., Volume 26, Number 4 (1955), 664-685.

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177728427

Digital Object Identifier
doi:10.1214/aoms/1177728427

Mathematical Reviews number (MathSciNet)
MR73889

Zentralblatt MATH identifier
0066.38603

JSTOR