Tests based on sum-functions of spacings for uniform random numbers

We examine the idea of testing uniform random number generators via two goodness-of-fit statistics: the sum of the logarithms and the sum of squares of overlapping m-spacings. The first statistic is related to an estimator of the entropy of a density and is good to detect clustering, whereas the second one, known as Greenwood's statistic for m = 1, is optimal in terms of Pitman efficiency, in certain setups, among sum-functions of m-spacings. These statistics are asymptotically normally distributed. We evaluate the distance between the standard normal distribution and that of the standardized statistics, as a function of m and of the sample size, when standardization is done using either the asymptotic or the exact (for finite sample size) mean and variance. We then report on experiments with these statistics to detect defects in some popular random number generators.