The Precision of Unbiased Ratio-Type Estimators

Abstract When each unit in a population of N units consists of an x and a y measurement, where the population mean Xbar of the x's is known, then the population mean Ybar of the y's is frequently estimated by drawing a random sample of n units and using one of the customary biased ratio-type estimators, Ybar (based on the ratio of the y sample mean and the x sample mean), or Ybar (based on the sample mean of the ratios of the y and x measurements for each unit). The precision of these estimators is compared with an exact formula derived here (when N>>n) for the variance of an unbiased ratio-type estimator yǐ, a modification of Ybar and Ybar, introduced in 5. Conditions pertaining to the population moments are obtained under which Var yǐ < Var Ybar, and it is also shown that Var yǐ < Var Ybar (when nŕ ∞) if and only if the slope of the population regression line of y on x is closer to the population mean of the ratios of the y and x measurements for each unit than to Ybar/Xbar. The estimation of the variances of the three ratio-type estimators is discussed, and some numerical examples are presented to illustrate the comparison of the estimators. Certain special populations are examined in detail, and the relation between regression and ratio-type estimators is also discussed.