Multi-Domain Neyman-Tchuprov Optimal Allocation

Abstract

The eigenproblem solution of the multi-domain efficient allocation is identified as a direct generalization of the classical Neyman-Tchuprov optimal allocation in stratified SRSWOR. This is achieved through analysis of eigenvalues and eigenvectors of a suitable population-based matrix D. Such a solution is an analytical companion to NLP approaches, which are often used in applications, see, e.g. Choudhry, Rao and Hidiroglou (2012). In this paper we are interested rather in the structure of the optimal allocation vector and relative variance than in such purely numerical tools (although the eigenproblem solution provides also numerical solutions, see, e.g. Wesołowski and Wieczorkowski (2017)). The domain-wise optimal allocation and the respective optimal variance of the estimator are determined by the unique direction (defined in terms of the positive eigenvector of matrix D) in the space R I , where I is the number of domains in the population.

Rotation schemes and Chebyshev polynomials

“…Under such a setting a general solution for arbitrary rotation pattern was obtained in Kowalski and Wesołowski (2015) (referred to by KW in the sequel). According to the main result in KW the recursion depth, p, is the size of the maximal gap in the rotation pattern increased by 1 (therefore it was 1 in the Patterson model, 2 in for rotation patterns with gaps of size 1) and 3 in the LFS rotation pattern 110011 (the last one settled in Wesołowski (2010)). The form of the coefficients in (3), as given in KW, is explicit, and rather unexpectedly, involves the Chebyshev polynomials of the first kind defined by…”