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 the overlapping and the non-overlapping vectors are considered. For different families of generators, such as the 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 i.i.d. uniformity 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 k n (in contrast to what is usually done or recommended in simulation books) and use overlapping vectors.