arXiv subjects

Wojciech Wójciak

Publications and source records attributed to Wojciech Wójciak.

3 recordsLinked to original sources

arXiv2024

Recursive Neyman Algorithm for Optimum Sample Allocation under Box Constraints on Sample Sizes in Strata

The optimum sample allocation in stratified sampling is one of the basic issues of survey methodology. It is a procedure of dividing the overall sample size into strata sample sizes in such a way that for given sampling designs in strata the variance of the stratified $π$ estimator of the population total (or mean) for a given study variable assumes its minimum. In this work, we consider the optimum allocation of a sample, under lower and upper bounds imposed jointly on sample sizes in strata. We are concerned with the variance function of some generic form that, in particular, covers the case of the simple random sampling without replacement in strata. The goal of this paper is twofold. First, we establish (using the Karush-Kuhn-Tucker conditions) a generic form of the optimal solution, the so-called optimality conditions. Second, based on the established optimality conditions, we derive an efficient recursive algorithm, named RNABOX, which solves the allocation problem under study. The RNABOX can be viewed as a generalization of the classical recursive Neyman allocation algorithm, a popular tool for optimum allocation when only upper bounds are imposed on sample strata-sizes. We implement RNABOX in R as a part of our package stratallo which is available from the Comprehensive R Archive Network (CRAN) repository.

stat.ME↗

arXiv2023

Another Solution for Some Optimum Allocation Problem

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.

math.ST↗

arXiv2021

Optimality of the recursive Neyman allocation

We derive a formula for the optimal sample allocation in a general stratified scheme under upper bounds on the sample strata-sizes. Such a general scheme includes SRSWOR within strata as a special case. The solution is given in terms of V-allocation with V being the set of take-all strata. We use V-allocation to give a formal proof of optimality of the popular recursive Neyman algorithm, rNa. This approach is convenient also for a quick proof of optimality of the algorithm of Stenger and Gabler (2005), SGa, as well as of its modification, coma, we propose here. Finally, we compare running times of rNa, SGa and coma. Ready-to-use R-implementations of these algorithms are available on CRAN repository at https://cran.r-project.org/web/packages/stratallo.

math.ST↗