Uniform point sets and the collision test

Monte Carlo and quasi-Monte Carlo methods are popular numerical tools used in many applications. The quality of the pseudorandom sequence used in a Monte Carlo simulation is essential to the accuracy of its estimates. Likewise, the quality of the low-discrepancy sequence determines the accuracy of a quasi-Monte Carlo simulation. There is a vast literature on statistical tests that help us assess the quality of a pseudorandom sequence. However, for low-discrepancy sequences, assessing quality by estimating discrepancy is a very challenging problem, leaving us with no practical options in very high dimensions. In this paper, we will discuss how a certain interpretation of the well-known collision test for pseudorandom sequences can be used to obtain useful information about the quality of low-discrepancy sequences. Numerical examples will be used to illustrate the applications of the collision test.