Statement of Contribution Close-point Spatial Tests and Their Application to Random Number Generators Close-point Spatial Tests and Their Application to Random Number Generators

Statement Ideally, a random number generator used for a given simulation problem should pass statistical tests closely related to that problem. But tailoring special tests for each problem is impractical. General purpose generators must then be submitted to a wide range of varied statistical tests, sensitive to diierent types of defects in the generator. In this article, we study statistical tests based on nearest-neighbor distances between the points formed by vectors of uniform random numbers. Such tests are relevant and necessary, because Monte Carlo simulation is often used to estimate the (complicated) distribution of certain statistics related to nearest-neighbor distances. The contribution of this paper is to provide an empirical study of these spatial tests from three viewpoints. Firstly, the distributions of most of these test statistics are known only asymptotically, and we evaluate, by simulation, the approximation error of the exact distribution by its asymptotic counterpart. Secondly, we investigate practical implementation issues for the tests. Thirdly, we study how certain classes of commonly used random number generators behave with respect to these tests. Linear congruential generators fail spectacularly when the sample size gets close to the square root of the period length. Since a sample size of 2 16 points is small and easy to handle for today's computers, one should conclude that linear congruential generators with period length around 2 32 , which are still commonly used, are inappropriate. We study statistical tests of uniformity based on the L p-distances between the m nearest pairs of points, for n points generated uniformly over the k-dimensional unit hypercube or unit torus. The number of distinct pairs at distance no more than t, for t 0, is a stochastic process whose initial part, after an appropriate transformation and as n ! 1, is asymptotically a Poisson process with unit rate. Convergence to this asymptotic is slow in the hypercube as soon as k exceeds 2 or 3, due to edge eeects, but is reasonably fast in the torus. We look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions, discuss computational issues, and apply the tests to random number generators. Linear congruential generators fail decisively certain variants of the tests as soon as n approaches the square root of the period length.