An eigenproblem approach to optimal equal-precision sample allocation in subpopulations

TL;DR

This paper introduces an eigenproblem-based method for optimal equal-precision sample allocation in two-stage sampling, providing a simple, computationally efficient solution that ensures uniform precision across subpopulations.

Contribution

It presents a novel eigenvalue approach to solve the equal-precision allocation problem in two-stage sampling, simplifying calculations and accommodating different precision priorities.

Findings

Method is based on eigenvalues and eigenvectors of matrices derived from population data.

Solution is simple, universal, and implementable with standard linear algebra software.

Illustrated with a real-world Labour Force Survey example.

Abstract

Allocation of samples in stratified and/or multistage sampling is one of the central issues of sampling theory. In a survey of a population often the constraints for precision of estimators of subpopulations parameters have to be taken care of during the allocation of the sample. Such issues are often solved with mathematical programming procedures. In many situations it is desirable to allocate the sample, in a way which forces the precision of estimates at the subpopulations level to be both: optimal and identical, while the constraints of the total (expected) size of the sample (or samples, in two-stage sampling) are imposed. Here our main concern is related to two-stage sampling schemes. We show that such problem in a wide class of sampling plans has an elegant mathematical and computational solution. This is done due to a suitable definition of the optimization problem, which enables to solve it through a linear algebra setting involving eigenvalues and eigenvectors of matrices defined in terms of some population quantities. As a final result we obtain a very simple and relatively universal method for calculating the subpopulation optimal and equal-precision allocation which is based on one of the most standard algorithms of linear algebra (available e.g. in R software). Theoretical solutions are illustrated through a numerical example based on the Labour Force Survey. Finally, we would like to stress that the method we describe, allows to accommodate quite automatically for different levels of precision priority for subpopulations.