Symmetry property of multiplicative congruential random number generator in chi-square test

Abstract

The k-term prime modulus multiplicative congruential generator: Rn = (a1Rn-1 +...,+ akRn-k ) mod m, is able to produce numbers (RNs) of full period mk-1 when the multipliers a1....,ak are chosen properly. In testing uniformity, the full period of RNs is usually divided into segments to calculate the chi-square statistics of the segments and test subsequently whether these statistics conform to a chi-square distribution. A symmetry property is that if an even number of segments, say 2s, is divided, then the chi-square statistic calculated from the ith segment of the first s segments is the same as that of the ith segment of the last s segments. Based on this property, the computational effort usually needed in testing uniformity is reduced by half.