On the Efficiency of Raking Ratio Estimation for Multiple Frame Surveys

Multiple frame surveys involve the combination of samples selected from separate sampling frames. A topical example is a dual-frame survey that combines a sample of households interviewed by telephone with a sample drawn from a frame including nontelephone households. Various estimators of population totals and means have been proposed for multiple frame surveys. In particular Bankier proposed the application of raking ratio estimation and compared its performance to alternative estimators in a numerical study based on dual-frame Statistics Canada data. He concluded that the raking ratio estimator performed well but gave no theoretical results to support his empirical findings. This article provides a theoretical study of the efficiency of the raking ratio estimator for multiple-frame surveys. Attention is restricted mainly to the classical, unstratified two-frame case considered by Hartley. For the estimation of totals, we give a closed-form expression for the limiting form of the raking ratio estimator as the number of iterations increases. We show that this limiting estimator is consistent and obtain an expression for its asymptotic variance. Using this expression we show that this estimator is uniformly less efficient than an estimator proposed by Fuller and Burmeister. However, the loss of efficiency is small in some special cases. The raking ratio estimator has the advantage that it extends simply to more than two frames and to stratified sampling, whereas the corresponding extension of the Fuller–Burmeister estimator is more problematic.