Survey Methodology
Constructing all determinantal sampling designs

by Vincent Loonis^{Note 1}

Abstract

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 $P^{\Pi }$ 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 $P^{\Pi }.$ 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.

Key Words: Determinantal process; Balanced sampling; Semidefinite optimization.

Table of contents

• Section 1. Introduction
• Section 2. Algebraic notations and reminders
• Section 3. Reminders about determinantal sampling designs
• Section 4. Constructing all determinantal sampling designs with fixed first-order inclusion probabilities
• Section 5. Matrices $K_N^{{\Pi }^{⊳}}\mathrm{<mo\; stretchy="false">(</mo>}M\mathrm{,}\Omega \mathrm{,}\rho \mathrm{<mo\; stretchy="false">)</mo>}$ for specific values of $\Omega$
• Section 6. Applications
• Section 7. Conclusion
• Acknowledgements
• Appendix
• References

How to cite

Loonis, V. (2023). Constructing all determinantal sampling designs. Survey Methodology, Statistics Canada, Catalogue No. 12‑001-X, Vol. 49, No. 2. Paper available at http://www.statcan.gc.ca/pub/12-001-x/2023002/article/00008-eng.htm.

Note

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