Neyman-type sample allocation for domains-efficient estimation in multistage sampling

Abstract

We consider a problem of allocation of a sample in two-and three-stage sampling. We seek allocation which is both multi-domain and population efficient. Choudhry et al. (Survey Methods 38(1):23-29, 2012) recently considered such problem for one-stage stratified simple random sampling without replacement in domains. Their approach was through minimization of the sample size under constraints on relative variances in all domains and on the overall relative variance. To attain this goal, they used nonlinear programming. Alternatively, we minimize here the relative variances in all domains (controlling them through given priority weights) as well as the overall relative variance under constraints imposed on total (expected) cost. We consider several two-and three-stage sampling schemes. Our aim is to shed some light on the analytic structure of solutions rather than in deriving a purely numerical tool for sample allocation. To this end, we develop the eigenproblem methodology introduced in optimal allocation problems in Niemiro and Wesołowski (Appl Math 28:73-82, 2001) and recently updated in Wesołowski and Wieczorkowski (Commun Stat Theory Methods 46(5):2212-2231, 2017) by taking under account several new sampling schemes and, more importantly, by the (single) total expected variable cost constraint. Such approach allows for solutions which are direct generalization of the Neyman-type allocation. The structure of the solution is deciphered from the explicit allocation formulas given in terms of an eigenvector v * of a population-based matrix D. The solution we provide can be viewed as a multi-domain version of the Neyman-type allocation in multistage stratified SRSWOR schemes.

Multi-Domain Neyman-Tchuprov Optimal Allocation

“…This is the essence of the present paper, in which we describe the eigenproblem approach as a generalization of the classical Neyman-Tchuprov methodology to the case of multi-domain optimal allo-cation. Such eigenproblem setting in the context of the domain-wise efficient allocation originally was proposed in Niemiro and Wesołowski (2001), and developed more recently in Wesołowski and Wieczorkowski (2017), and Khan and Wesołowski (2019). In the first of these papers the authors considered two-stage sampling schemes with SRSWOR and stratification either at the first or at the second stage.…”