On the dependence structure and quality of scrambled (t, m, s)-nets

Abstract

In this paper we develop a framework to study the dependence structure of scrambled {(t,m,s)}-nets. It relies on values denoted by {C_{b}({\boldsymbol{k}};P_{n})}, which are related to how many distinct pairs of points from {P_{n}} lie in the same elementary {{\boldsymbol{k}}}-interval in base b. These values quantify the equidistribution properties of {P_{n}} in a more informative way than the parameter t. They also play a key role in determining if a scrambled set {\widetilde{P}_{n}} is negative lower orthant dependent (NLOD). Indeed, this property holds if and only if {C_{b}({\boldsymbol{k}};P_{n})\leq 1} for all {{\boldsymbol{k}}\in\mathbb{N}^{s}}, which in turn implies that a scrambled digital {(t,m,s)}-net in base b is NLOD if and only if {t=0}. Through numerical examples we demonstrate that these {C_{b}({\boldsymbol{k}};P_{n})} values are a powerful tool to compare the quality of different {(t,m,s)}-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.

Walsh functions, scrambled $(0,m,s)$-nets, and negative covariance: applying symbolic computation to quasi-Monte Carlo integration

“…This is based on the concept of negative dependence. Following this, it was shown in Wiart et al [14], that scrambled (0, m, s)-nets do no worse for functions that are "quasi-monotone". They did this by showing that scrambled (0, m, s)-nets are negative lower orthant dependent which allowed them to apply a previous result by Lemieux [6].…”

“…where As in [14], a scrambled (t, m, s)-net in base b is a (t, m, s)-net that has been digitally scrambled in base b (see Definition 3).…”

“…Theorem 4. (Wiart et al [14]) Let Pn be a scrambled (t, m, s)-net in base b whose one-dimensional projections are (0, m, 1)-nets. Then the joint pdf ψ(x, y) of two distinct points randomly chosen from Pn is given by…”

“…for all i ∈ N s with i being the sum of the coordinates of i [14]. Since the t parameter tells us nothing about the distribution of a (t, m, s)-net P n on elementary k−intervals where…”

“…One aspect of our work in the present paper that differs from the recent work of Wiart et al [14] is that we take the framework that has already been fixed and investigate the integration of L 2 ([0, 1) s ) functions with Walsh decompositions over the scrambled (0, m, s)-nets. Elementary Walsh functions are piecewise constant and form an orthonormal basis for L 2 ([0, 1) s ).…”