In Search of Correlation in Multiplicative Congruential Generators with Modulus 2³¹ -1

Abstract

This paper describes an empirical search for correlation in sample sequences produced by 16 multiplicative congruential random number generators with modulus 2³¹ - 1. Each generator has a distinct multiplier. One multiplier is in common use in the LLRANDOM and IMSL random generation packages as well as in APL and SIMPL/1. A second is used in SIMSCRIPT II. Six multipliers were taken from a recent study that showed them to have the best spectral and lattice test properties among 50 multipliers considered. The last eight multipliers had the poorest spectral and lattice test properties for 2-tupes among the 50. A well known poor generator, RANDU, with modulus 2³¹, was also tested to provide a benchmark for evaluating the empirical testing procedure.

A comprehensive analysis based on test statistics derived from cumulative periodograms computed for each multiplier for each of 512 independent replications of 16384 observations each showed evidence of excess high frequency variation in two multipliers and excess midrange frequency variation in three others, including RANDU. Also evidence exists for a bimodal spectral density function for yet another multiplier. An examination of the test results showed that the empirical evidence of a departure from independence did not significantly favor the eight poorest multipliers. This observation is in agreement with a similar observation made by the authors in an earlier study of these multipliers that principally concentrated on their distributional properties in one, two and three dimensions. This consistency raises some doubt as to how one should interpret the results of the spectral and lattice tests for a multiplier. Also, the three multipliers considered superior in the earlier study maintain that position in the current study.