The Precision of Some Unbiased Regression Estimators

Abstract

In the development of the statistical theory of sample surveys, one of the main aims has been to improve the estimates that are obtained from the data. One method which has been used quite extensively to create estimators with more advantageous variances is the utilization of auxiliary information. Analytically this usually takes the form of a random sample of n pairs (yi , xi) from a population of size N. The problem is to estimate the population y mean (guy) relative to the assumption that the population x mean (Ax) is known exactly. This information can be utilized in the form of ratio and regression estimators. The distinction between ratio and regression estimators is that a regression estimator is invariant under linear x transformations and undergoes the same linear transformation as y. Ratio estimators have the same property but only for scale changes. Both types of estimator result in a comparison of the over-all variation in the y characteristic to the residual variation about the relationship of y and x. Ratio estimators are inefficient if the relationship between y and x does not pass through the origin; regression estimators do not require this. Descriptions can be found, for example, in Cochran [1953]. The usual regression estimator assumes a linear model,