On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets

TL;DR

This paper investigates the negative dependence properties of Latin hypercube samples and scrambled nets, introducing the correlation number concept, and shows these samples are not fully negatively orthant dependent but still exhibit useful negative dependence properties.

Contribution

It introduces the correlation number to quantify negative dependence and analyzes the dependence properties of Latin hypercube sampling and $(t,m,d)$-nets, revealing they are not fully negatively orthant dependent.

Findings

Latin hypercube samples are not negatively orthant dependent.

Latin hypercube sampling in dimension d has a gamma ≤ e^d.

Discrepancy bounds depend mildly on the gamma value.

Abstract

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.