Another Solution for Some Optimum Allocation Problem

TL;DR

This paper introduces a new optimality condition-based method, LRNA, for stratified sampling allocation problems, providing a formal proof of its optimality and a practical R implementation.

Contribution

It formulates and proves the optimality of the LRNA procedure for sample allocation with bounds, extending classical methods like Neyman allocation.

Findings

LRNA solves the allocation problem with bounds effectively.

The method is proven optimal using Karush-Kuhn-Tucker conditions.

An R package implementation is available on CRAN.

Abstract

We derive optimality conditions for the optimum sample allocation problem in stratified sampling, formulated as the determination of the fixed strata sample sizes that minimize the total cost of the survey, under the assumed level of variance of the stratified $\pi$ estimator of the population total (or mean) and one-sided upper bounds imposed on sample sizes in strata. In this context, we presume that the variance function is of some generic form that, in particular, covers the case of the simple random sampling without replacement design in strata. The optimality conditions mentioned above will be derived from the Karush-Kuhn-Tucker conditions. Based on the established optimality conditions, we provide a formal proof of the optimality of the existing procedure, termed here as LRNA, which solves the allocation problem considered. We formulate the LRNA in such a way that it also provides the solution to the classical optimum allocation problem (i.e. minimization of the estimator's variance under a fixed total cost) under one-sided lower bounds imposed on sample sizes in strata. In this context, the LRNA can be considered as a counterparty to the popular recursive Neyman allocation procedure that is used to solve the classical problem of an optimum sample allocation with added one-sided upper bounds. Ready-to-use R-implementation of the LRNA is available through our stratallo package, which is published on the Comprehensive R Archive Network (CRAN) package repository.