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The Discrete Runs Test and the Discrete Maximum of *t* Test

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Technical Report CS 96-15

Andrew Shapira

ECSE Department

Rensselaer Polytechnic Institute

Troy, NY 12180

June 10, 1996

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Abstract

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 *{ 0, 1, ..., d-1 }*, for some integer
*d > 1*.
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*.
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Key Words

Runs test,
maximum of *t* test,
random number testing,
random number generation,
chi-square test,
empirical testing.

Questions/Comments: shapiraa@cs.rpi.edu