Minimum Sample Size Allocation in Stratified Sampling Under Constraints on Variance and Strata Sample Sizes
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We derive optimality conditions for the optimal sample allocation problem, formulated as the determination of the fixed strata sample sizes that minimize total sample size, under assumed level of the variance of the stratified π-estimator and one-sided upper bounds imposed on strata sample sizes. In this paper, this problem is considered in the context of general stratified sampling scheme that includes simple random sampling without replacement design within strata as a special case. Based on established optimality conditions, we create a new algorithm, the LrNa, that solves the allocation problem defined above. This new algorithm has its origin in popular recursive Neyman allocation procedure, the rNa, that is used to solve classical optimal sample allocation problem (i.e. minimization of the π-estimator's variance under fixed total sample size) with only one-sided upper bounds constraints imposed on strata sample sizes (see e.g. Särndal, Swensson, and Wretman (1992, Remark 12.7.1, p. 466), or Wesołowski, Wieczorkowski, and Wójciak (2021)). Ready-to-use R-implementation of the LrNa is available on CRAN repository at https://cran.r-project.org/web/packages/stratallo.
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