Richard J. Simard

TestU01: A C library for empirical testing of random number generators

We introduce TestU01, a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generators period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01, the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.

On the performance of birthday spacings tests with certain families of random number generators

We examine how a statistical test based on discrete spacings between points, in one or more dimensions, detects the regularities in certain popular classes of random number generators. We provide a rule of thumb giving the minimal sample size for the test to reject the generator systematically, as a function of the generator’s size (or period length), for generator families such as the linear congruential, Taus worthe, non linear inversive, etc. Full period linear congruential generators with a good behavior in the spectral test, for example, start to fail the two-dimensional test decisively at sample sizes approximately equal to the cubic root of their period length (or modulus).

Sparse Serial Tests of Uniformity for Random Number Generators

Different versions of the serial test for testing the uniformity and independence of vectors of successive values produced by a (pseudo)random number generator are studied. These tests partition the t-dimensional unit hypercube into k cubic cells of equal volume, generate n points (vectors) in this hypercube, count how many points fall in each cell, and compute a test statistic defined as the sum of values of some univariate function f applied to these k individual counters. Both overlapping and nonoverlapping vectors are considered. For different families of generators, such as linear congruential, Tausworthe, nonlinear inversive, etc., different ways of choosing these functions and of choosing k are compared, and formulas are obtained for the (estimated) sample size required to reject the null hypothesis of independent uniform random variables, as a function of the period length of the generator. For the classes of alternatives that correspond to linear generators, the most efficient tests turn out to have

Beware of linear congruential generators with multipliers of the form a = ±2 q ±2 r

k \gg n

Close-Point Spatial Tests and Their Application to Random Number Generators

(in contrast to what is usually done or recommended in simulation books) and to use overlapping vectors.

Static Network Reliability Estimation via Generalized Splitting

Linear congruential random-number generators with Mersenne prime modulus and multipliers of the form <italic>a</italic> = ±2<italic><supscrpt>q</supscrpt></italic> ±<italic><supscrpt>r</supscrpt></italic> have been proposed recently. Their main advantage is the availability of a simple and fast implementation algorithm for such multipliers. This note generalizes this algorithm, points out statistical weaknesses of these multipliers when used in a straightforward manner, and suggests in what context they could be used safely.

Random Numbers for Parallel Computers: Requirements and Methods, With Emphasis on GPUs

We study statistical tests of uniformity based on theL p -distances between them nearest pairs of points, forn points generated uniformly over thek-dimensional unit hypercube or unit torus. The number of distinct pairs at distance no more thant, fort = 0, is a stochastic process whose initial part, after an appropriate transformation and asn ? 8, is asymptotically a Poisson process with unit rate. Convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3, due to edge effects, 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 asn approaches the square root of the period length.

Inverting the symmetrical beta distribution

We propose a novel simulation-based method that exploits a generalized splitting GS algorithm to estimate the reliability of a graph or network, defined here as the probability that a given set of nodes are connected, when each link of the graph fails with a given small probability. For large graphs, in general, computing the exact reliability is an intractable problem and estimating it by standard Monte Carlo methods poses serious difficulties, because the unreliability one minus the reliability is often a rare-event probability. We show that the proposed GS algorithm can accurately estimate extremely small unreliabilities and we exhibit large examples where it performs much better than existing approaches. It is also flexible enough to dispense with the frequently made assumption of independent edge failures.

On the Lattice Structure of a Special Class of Multiple Recursive Random Number Generators

We examine the requirements and the available methods and software to provide (or imitate) uniform random numbers in parallel computing environments. In this context, for the great majority of applications, independent streams of random numbers are required, each being computed on a single processing element at a time. Sometimes, thousands or even millions of such streams are needed. We explain how they can be produced and managed. We devote particular attention to multiple streams for GPU devices.

Static Network Reliability Estimation under the Marshall-Olkin Copula

We propose a fast algorithm for computing the inverse symmetrical beta distribution. Four series (two around x = 0 and two around x = 1/2) are used to approximate the distribution function, and its inverse is found via Newtons method. This algorithm can be used to generate beta random variates by inversion and is much faster than currently available general inversion methods for the beta distribution. It turns out to be very useful for generating gamma processes efficiently via bridge sampling.