The Discrete Runs Test and the Discrete Maximum of T Test

Empirical tests are used to help determine whether a given sequence of numbers has statistical properties similar to those of a sequence whose elements are drawn from a uniform probability distribution. The runs test and the maximum of t test are two traditional tests that have been applied to sequences of real numbers between 0 and 1. Motivated by the desire for a set of empirical tests that all follow the pattern of taking n measurements and performing a chi-square test on the n measurements, this report examines discrete versions of both tests that are applied directly to any sequence whose elements are drawn from f0; 1; : : : ; d ? 1g, for some integer d 2. The tests are practical and do not involve converting integers to a pseudo-continuous number through division by d. Expressions are derived for the probabilities needed to carry out the discrete tests. The utility of the discrete tests is compared to that of the continuous versions. When the tests are considered alone, and not within a collection of other tests, the discrete runs test seems preferable to the continuous version for all values of d, whereas the discrete maximum of t test appears to be preferable to the continuous version mainly for small values of d.