Another solution for some optimum allocation problem

Wojciech Wójciak

doi:https://doi.org/10.59170/stattrans-2023-071

Another solution for some optimum allocation problem

Wojciech Wójciak Warsaw University of Technology, Poland ORCID:https://orcid.org/0000-0002-5042-160X Statistics in Transition new series, vol. 24, 2023, 5, pages: 203-219 Published online: 7 December 2023 https://doi.org/10.59170/stattrans-2023-071 Citation: Wójciak, W., 2023. Another solution for some optimum allocation problem. Statistics in Transition new series, 24(5), pp. 203-219. https://doi.org/10.59170/stattrans-2023-071

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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 𝜋 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.

KEYWORDS

stratified sampling, optimum allocation, minimum cost allocation under upper bounds, optimum allocation constant variance, optimum allocation under lower bounds, recursive Neyman algorithm

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