Tuning the Collision Test for Stringency

The collision test is one of the most important statistical tests for random number generators. It simulates the throwing of balls randomly into urns. A problem in applying this test is to determine the number of urns, m, and the number of balls, n, so that the test is among the most stringent possible on a computer available for testing. Our studies showed that for a fixed m, n shall be determined subject to maximizing the variance of the number of collisions. With this rule, the stringency of the resulting test increases as m increases. And the test of 23 2 ≥ m flunked a recorded number of generators, including congruential generators, shift-register generators, additive generators of lags less than 40, subtract-with-borrow generators of lags less than 24, and a combination of a congruential and a shift-register generator.