Lower Bounds of Unanchored Discrepancy for Mixed-Level U-Type Designs
Keywords:
Uniform design, Unanchored discrepancy, Lower bound, U-type design, Threshold acceptingAbstract
Uniform $U$-type designs are fundamental to modern experimental methodology, particularly in computer experiments and robust parameter design, owing to their ability to uniformly distribute points across the design space. This space-filling property confers resilience against model misspecification and provides an ideal foundation for nonparametric estimation. Despite their practical significance, constructing uniform designs that minimize a given discrepancy poses an inherently intractable combinatorial optimization problem. This challenge has motivated the pursuit of theoretical lower bounds, which serve as optimality benchmarks and termination criteria for heuristic search algorithms. Mixed two- and three-level designs are ubiquitous in practical experimental settings, including industrial experiments, clinical trials, and computer simulations. In this paper, we derive sharp, analytically tractable lower bounds for the unanchored discrepancy in mixed two- and three-level U-type designs. Leveraging Jensen's inequality and combinatorial optimization, we obtain closed-form expressions that are both computationally expedient and demonstrably tight. We further delineate necessary conditions under which these bounds are attainable and propose constructive methodologies for generating designs that achieve or approach these theoretical limits. The resulting benchmarks fill a critical gap in the theory of uniform designs and offer tangible utility in constructing optimal experimental plans for computer simulations and robust parameter studies.
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