Goodness-of-fit tests in many dimensions

A method is presented to construct goodness-of-fit statistics in many dimensions for which the distribution of all possible test results in the limit of an infinite number of data becomes Gaussian if also the number of dimensions becomes infinite. Furthermore, an explicit example is presented, for which this distribution as good as only depends on the expectation value and the variance of the statistic for any dimension larger than one.Comment: 14 page

The dynamical point of view of low-discrepancy sequences

International audienceIn this overview we show by examples, how to associate certain sequences in the higher-dimensional unit cube to suitable dynamical systems. We present methods and notions from ergodic theory that serve as tools for the study of low-discrepancy sequences and discuss an important technique, cutting- and-stacking of intervals

Quantum field theory for discrepancies

The concept of discrepancy plays an important role in the study of uniformity properties of point sets. For sets of random points, the discrepancy is a random variable. We apply techniques from quantum field theory to translate the problem of calculating the probability density of (quadratic) discrepancies into that of evaluating certain path integrals. Both their perturbative and non-perturbative properties are discussed.Comment: 26 page

Gaussian limits for discrepancies. I: Asymptotic results

We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit $N\to\infty$. We then examine the circumstances under which this distribution approaches a normal distribution. For large classes of non-uniformity measures, a Law of Many Modes in the spirit of the Central Limit Theorem can be derived.Comment: 25 pages, Latex, uses fleqn.sty, a4wide.sty, amsmath.st

On the use of a Modified Latin Hypercube Sampling (MLHS) approach in the estimation of a Mixed Logit model for vehicle choice

Quasi-random number sequences have been used extensively for many years in the simulation of integrals that do not have a closed-form expression, such as Mixed Logit and Multinomial Probit choice probabilities. Halton sequences are one example of such quasi-random number sequences, and various types of Halton sequences, including standard, scrambled, and shuffled versions, have been proposed and tested in the context of travel demand modeling. In this paper, we propose an alternative to Halton sequences, based on an adapted version of Latin Hypercube Sampling. These alternative sequences, like scrambled and shuffled Halton sequences, avoid the undesirable correlation patterns that arise in standard Halton sequences. However, they are easier to create than scrambled or shuffled Halton sequences. They also provide more uniform coverage in each dimension than any of the Halton sequences. A detailed analysis, using a 16-dimensional Mixed Logit model for choice between alternative-fuelled vehicles in California, was conducted to compare the performance of the different types of draws. The analysis shows that, in this application, the Modified Latin Hypercube Sampling (MLHS) outperforms each type of Halton sequence. This greater accuracy combined with the greater simplicity make the MLHS method an appealing approach for simulation of travel demand models and simulation-based models in general