Two-stage indirect determinantal sampling designs

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.