Dependence properties of scrambled Halton sequences

Abstract

In this paper, scrambled Halton sequences are shown to have a form of negative dependence that is desirable for the purpose of improving upon the Monte Carlo method for multivariate integration. The scrambling methods with these properties are based on either the nested uniform permutations of Owen or the random linear scrambling of Matoušek. The framework of negative dependence is also used to develop new criteria for assessing the quality of generalized Halton sequences, in such a way that they can be analyzed for finite (potentially small) point set sizes and be compared to digital net constructions. Using this type of criteria, parameters for a new generalized Halton sequence are derived. Numerical results are presented to compare different generalized Halton sequences and their randomizations.