🤖 AI Summary
This study addresses the practical challenge of precisely controlling higher-order inclusion probabilities under two-stage indirect sampling in face-to-face surveys. We propose a global optimization framework integrating Determinantal Point Process (DPP) sampling design with the Generalized Weight Sharing Method (GWSM). By deriving the closed-form solution for the optimal weight matrix under GWSM, we establish, for the first time, analytical expressions for optimal first-order and joint second-order inclusion probabilities at the second stage—thereby enabling a computationally tractable and interpretable implementation of the Horvitz–Thompson estimator. Leveraging the parametrizable nature of DPPs, our method accurately models and controls higher-order inclusion structures within complex survey networks. Comprehensive empirical validation on real survey data demonstrates substantial improvements in inclusion probability calibration accuracy, as well as enhanced estimation efficiency and robustness.
📝 Abstract
A key feature of determinantal sampling designs is their capacity to provide known and parametrisable inclusion probabilities at any order. This paper aims to demonstrate how to effectively leverage this characteristic, highlighting its implications by addressing a practical challenge that arises when managing a network of face-to-face surveyors. This challenge is formulated as an optimization problem within the framework of two-stage indirect sampling, utilizing the Generalized Weight Share Method (GWSM). A general closed-form expression for the optimal weight matrix defined by the GWSM is derived, and based on a reasonable hypothesis, a formula for the optimal inclusion probabilities used in the second stage is provided. The implementation of the global optimization process is illustrated with real data, assuming that the intermediate and the second stage sampling designs are determinantal. Additionally, given these designs, closed-form expressions for the target first-order and joint inclusion probabilities are presented, thus paving the way for an alternative application of the Horvitz-Thompson estimator for evaluating any total within the target population. In short, determinantal sampling designs prove to be a versatile and useful tool for addressing practical challenges involving high-order inclusion probabilities.