Optimal allocation in stratified sampling with a nonlinear cost function
Let a population of \(N\) units be divided into \(L\) subpopulations of \(N_1,N_2,\dots,N_L\) units and let \(n_k\) be the number of sampling units selected from stratum \(k\). Let \(C=\sum\limits_{k=1}^L{c_k f(n_k)}\) be a cost function of the sampling, where \(c_k \) is the cost per unit in the \(k\)-th stratum and \(f\) is a known function. The problem is to minimize the cost function \(C\). In the case of the linear cost function \((f(x)=x,\;C=\sum\limits_{k = 1}^L {c_k } n_k)\) the optimal allocation has been given by \textit{W.G. Cochran} [Sampling techniques. 3rd ed. (1977; Zbl 0353.62011)]. Many examples of nonlinear cost functions have been given by \textit{R.M. Groves} [Survey errors and survey costs. (1989)].NEWLINENEWLINENEWLINEIn this paper the problem of optimal allocation in stratified sampling is considered for the power and logarithmic cost functions. Optimal allocations are obtained for sample sizes \(n_k \) in stratified sampling, for primary units and second-stage units in two-stage stratified sampling and for the size \(n'\) of the first sample and the share \(\nu_k \) of the second sample in stratum \(k\) which minimize cost for a fixed value of variance of estimate or minimizes variance for a fixed cost. The Lagrange multipliers method is used to solve the problems.
scientific article; zbMATH DE number 1631960