Abstract
We introduce a class of γ-negatively dependent random samples. We prove that this class includes, apart from Monte Carlo samples, in particular Latin hypercube samples and Latin hypercube samples padded by Monte Carlo.
For a γ-negatively dependent N-point sample in dimension d we provide probabilistic upper bounds for its star discrepancy with explicitly stated dependence on N, d, and γ. These bounds generalize the probabilistic bounds for Monte Carlo samples from Heinrich et al. (Acta Arith. 96 (2001) 279–302) and C. Aistleitner (J. Complexity 27 (2011) 531–540), and they are optimal for Monte Carlo and Latin hypercube samples. In the special case of Monte Carlo samples the constants that appear in our bounds improve substantially on the constants presented in the latter paper and in C. Aistleitner and M. T. Hofer (Math. Comp. 83 (2014) 1373–1381).
Acknowledgments
The authors thank Marcin Wnuk and two anonymous referees for valuable comments.
Part of the work of Michael Gnewuch was done while he was a research fellow and a visitor at the School of Mathematics and Statistics of the University of New South Wales in Sydney and a “Chercheur Invité” at the Laboratoire d’Informatique (LIX) of École Polytechnique in Paris, France. He acknowledges support from the Australian Research Council ARC and thanks his hosts Josef Dick, Frances Y. Kuo, Ian H. Sloan, and Benjamin Doerr for their hospitality.
Another part of his work was done while he visited special semesters and programs at the Radon Institute for Computational and Applied Mathematics (RICAM) in Linz, Austria, the Institute of Computational and Experimental Mathematics (ICERM) of Brown University in Providence, USA, and the Erwin Schrödinger Institute (ESI) in Vienna, Austria.
Citation
Michael Gnewuch. Nils Hebbinghaus. "Discrepancy bounds for a class of negatively dependent random points including Latin hypercube samples." Ann. Appl. Probab. 31 (4) 1944 - 1965, August 2021. https://doi.org/10.1214/20-AAP1638
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