Mixing of direct, ratio, and product method estimators

In a paper by SrivenkataramanaTracy[4], four methods of estimating a population total Ywith the use of an auxiliary variable were introduced, given a random sample without replacement from that population. These methods were “built around the idea that estimating the population total is essentially equivalent to estimating the total corresponding to the non‐sample units, since that corresponding to the sample units is known once the sample is drawn and measurements are made on it.” However, in the case of small sampling fractions the nonsample units constitute most of the population and no great improvement over the traditional estimators is to be expected. Therefore the methods are compared with the existing estimators and it turns out that they are special cases of the “mixing estimators”, introduced in this paper. The latter estimators can be made asymptotically equivalent to the regression estimator and are therefore asymptotically superior to all other estimators. An exact comparison is carried out on the artificial example given in [4]. The statement in this paper that “the proposed estimators are to be preferred to the regression estimator for., superiority of performance in the case of small samples” is evidently misleading. Finally a comparison is made with other “mixing‐type” estimators, that can prove very useful in practice.