Optimum Allocation of Stratified Random Samples Designed for Multiple Mean Estimates and Multiple Observed Variables

With a fixed overall sampling cost, it is essential to achieve minimal error variance of the estimated population statistics in order to be maximally efficient. The error variance of a population mean estimate derived from a stratified random sample (StRS) may be minimized by determining the optimum allocation of observations to the strata. A variety of approaches currently exists to achieve this. Moreover, for obvious economic reasons, a survey's objective is generally more than the estimation of a single population mean of a single variable. That is, mean values restricted to various subsets of strata and mean values belonging to several observed variables are simultaneously estimated. Available optimization procedures, however, only partially address this issue. In this paper, a method for optimum allocation is developed that explicitly allows the consideration of any number of observed variables and any number of mean estimates (as defined by the respective subsets of relevant strata). The proposed method is a generalization and unification of formulae provided by [Neyman, J. (1934 Neyman, J. 1934. On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection. J. Amer. Statist. Assoc., 33: 101–116. [Taylor & Francis Online] , [Google Scholar]). On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection. J. Amer. Statist. Assoc. 33:101–116, Tschuprow, A. A. (1923 Tschuprow, A. A. 1923. On the mathematical expectation of the moments of frequency distributions in the case of correlated observations. Metron, 2: 461–493. 646–683. [Google Scholar]). On the mathematical expectation of the moments of frequency distributions in the case of correlated observations. Metron 2:461–493, 646–683, Stuart, A. (1954 Stuart, A. 1954. A simple presentation of optimum sampling results. J. Roy. Stat. Soc., 33: 101–116. [Google Scholar]). A simple presentation of optimum sampling results. J. Roy. Stat. Soc. 33:101–116, and Yates, F. (1960 Yates, F. 1960. Sampling Methods for Censuses and Surveys London: Charles Griffin and Co. [Google Scholar]). Sampling Methods for Censuses and Surveys. London: Charles Griffin and Co.]. Examples demonstrate the usefulness of the proposed method. It is shown that the joint gain in the estimates' precision can be substantial.