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Pré-Publication, Document De Travail Année : 2021

On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets

Résumé

We study the notion of $\gamma$-negative dependence of random variables. This notion is a relaxation of the notion of negative orthant dependence (which corresponds to $1$-negative dependence), but nevertheless it still ensures concentration of measure and allows to use large deviation bounds of Chernoff-Hoeffding- or Bernstein-type. We study random variables based on random points $P$. These random variables appear naturally in the analysis of the discrepancy of $P$ or, equivalently, of a suitable worst-case integration error of the quasi-Monte Carlo cubature that uses the points in $P$ as integration nodes. We introduce the correlation number, which is the smallest possible value of $\gamma$ that ensures $\gamma$-negative dependence. We prove that the random variables of interest based on Latin hypercube sampling or on $(t,m,d)$-nets do, in general, not have a correlation number of $1$, i.e., they are not negative orthant dependent. But it is known that the random variables based on Latin hypercube sampling in dimension $d$ are actually $\gamma$-negatively dependent with $\gamma \le e^d$, and the resulting probabilistic discrepancy bounds do only mildly depend on the $\gamma$-value.
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Dates et versions

hal-03205337 , version 1 (22-04-2021)
hal-03205337 , version 2 (19-09-2021)

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  • HAL Id : hal-03205337 , version 1

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Benjamin Doerr, Michael Gnewuch. On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets. 2021. ⟨hal-03205337v1⟩
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