On Negatively Dependent Sampling Schemes, Variance Reduction, and Probabilistic Upper Discrepancy Bounds

Abstract

We study some notions of negative dependence of a sampling scheme that can be used to derive variance bounds for the corresponding estimator or discrepancy bounds for the underlying random point set that are at least as good as the corresponding bounds for plain Monte Carlo sampling. We provide new pre-asymptotic bounds with explicit constants for the star discrepancy and the weighted star discrepancy of sampling schemes that satisfy suitable negative dependence properties. Furthermore, we compare the different notions of negative dependence and give several examples of negatively dependent sampling schemes.

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• A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions
• On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets
• On the dependence structure and quality of scrambled $(t,m,s)-$nets
• The star discrepancy of a union of randomly digitally shifted Korobov polynomial lattice point sets depends polynomially on the dimension