Sample allocation in multivariate surveys - ARCHIVED

The optimum allocation to strata for multipurpose surveys is often solved in practice by establishing linear variance constraints and then using convex programming to minimize the survey cost. Using the Kuhn-Tucker theorem, this paper gives an expression for the resulting optimum allocation in terms of Lagrangian multipliers. Using this representation, the partial derivative of the cost function with respect to the k-th variance constraint is found to be -2 \alpha_{k^*} g (x^*) / v_k, where g (x^*) is the cost of the optimum allocation and where \alpha_{k^*} and v_k are, respectively, the k-th normalized Lagrangian multiplier and the upper bound on the precision of the k-th variable. Finally, a simple computing algorithm is presented and its convergence properties are discussed. The use of these results in sample design is demonstrated with data from a survey of commercial establishments.