Constructing all determinantal sampling designs

Description: In this article, we use a slightly simplified version of the method by Fickus, Mixon and Poteet (2013) to define a flexible parameterization of the kernels of determinantal sampling designs with fixed first-order inclusion probabilities. For specific values of the multidimensional parameter, we get back to a matrix from the family P^II from Loonis and Mary (2019). We speculate that, among the determinantal designs with fixed inclusion probabilities, the minimum variance of the Horvitz and Thompson estimator (1952) of a variable of interest is expressed relative to P^II. We provide experimental R programs that facilitate the appropriation of various concepts presented in the article, some of which are described as non-trivial by Fickus et al. (2013). A longer version of this article, including proofs and a more detailed presentation of the determinantal designs, is also available.

Issue Number: 2023002