Statistical Inference: A Short Course

Appendix 9.A Ratio Estimation

A procedure that is akin to that employed to estimate a population proportion p is ratio estimation. It can be used to determine a population ratio proper or to estimate a population mean. Ratio estimation involves taking measurements on two characteristics of interest of a sampling unit—one corresponding to the numerator and the other corresponding to the denominator of the ratio. In this regard, let a sample of size n consist of the n ordered pairs of observations (X_i, Y_i,), i = 1,. . .,n, where the sample values Y₁,. . .,Y_n refer to some measurable characteristic of the sampling unit that appears in the numerator of the ratio, and the sample values X₁,. . .,X_n pertain to some measurable characteristic of the sampling unit that appears in the denominator of the ratio. In what follows, the X_i's will serve as subsidiary information used for ratio estimation purposes.

At this point, some notation is in order. Let the population ratio appear as follows:

(9.A.1)

where the population totals for the Y- and X-characteristics are denoted as and , respectively, and and are the respective means for the said characteristics. Then an estimator for R is the sample ...