🤖 AI Summary
This study addresses the loss of sampling efficiency in stratified sampling under heteroscedastic populations, which arises when stratum variances are unknown and must be estimated. To tackle this issue, the authors propose a novel optimal sample allocation strategy grounded in Bayesian decision theory, specifically tailored for heteroscedastic univariate regression models. This approach extends Bayesian optimal experimental design to heteroscedastic settings for the first time, overcoming a key limitation of conventional Bayesian sampling designs that assume homoscedasticity and rely on suboptimal point estimates of design parameters. Empirical evaluations on both synthetic data and real-world IRS Form 990 charitable organization revenue data demonstrate that the proposed method consistently matches or outperforms established stratified sampling and model-assisted benchmark approaches.
📝 Abstract
We develop a Bayesian optimal sample allocation approach for stratified sampling in heteroscedastic populations. Existing optimal allocation theory typically assumes knowledge of certain design parameters (e.g., strata variances) that may be unknown, leading practitioners to substitute in survey-based estimates when planning samples, often without considering the effects of this substitution on sample efficiency. Bayesian decision theory for optimal experimental design avoids such substitutions and can be applied to sample allocation. Bayesian sample optimization methods were studied heavily from the mid-1960s through early 1980s, but have been overlooked since, in spite of modern computing advances that have facilitated a proliferation of Bayesian methods in other areas of statistics. A limitation of this early Bayesian sample design work is that it did not accommodate heteroscedastic error structures, which underlie commonly used ratio estimation models. Our paper, which optimizes the design under a univariate regression model with heteroscedastic errors, addresses this limitation of earlier work, while illustrating the Bayesian approach to design. We identify the optimal Bayesian allocation under our model, then compare performance of the proposed Bayesian sampling strategy with that of key design-based and model-assisted alternatives across several settings, finding that the proposed methods do as well or better than the alternatives under the scenarios considered. We apply our methods in analyzing revenues of public charities, using publicly available IRS Form 990 data.