SERIAL TEST
The serial (or digit) test based on digit expansion uses non-overlapping s-tuples of integers defined by the k-th to the k+l-1-th bit of every pseudorandom number. For a sample size n, the test is defined as follows: compute a chi-square test from n non-overlapping integer s-tuples by observing how often each of the possible
possible tuples occurs; repeat this m times; from the m corresponding chi-square values compute the KS statistic (denoted by t2(s,k,l) in the graphics below). A detailed description and empirical results for linear and inversive congruential generators are given in [1, 3, 4].
To illustrate some generators' behavior in the serial test, we present the
simulations produced by the ANSI C systemgenerator
the
excellent linear generator 
and by the
inversive congruential generator
for s=4, 
, 
and m = 64.
fishm is one of the best generators found by
Fishman and Moore [2] in an exhaustive search
which compared all maximum period generators 
with respect to their lattice structure in dimensions 
.
(1) ansi for s=4, and 
(in this and the following pictures, the values of 
are
truncated to 
to keep the graphics in scale; note
that 
):
(2) fishm for s=4, and :
(3) icg1 for s=4, and :
- 1
-
K. Entacher and H. Leeb.
Inversive pseudorandom number generators: empirical results.
In Proceedings of the Conference Parallel Numerics 95, Sorrento,
Italy, September 27-29, 1995, pages 15-27, 1995.
Abstract and postscript file available.
- 2
-
G.S. Fishman and L.R. Moore.
An exhaustive analysis of multiplicative congruential random number
generators with modulus
.
SIAM J. Sci. Statist. Comput., 7:24-45, 1986.
See erratum, ibid., 7:1058, 1986.
- 3
-
H. Leeb.
On the digit test.
In P. Hellekalek, G. Larcher, and P. Zinterhof, editors,
Proceedings of the 1st Salzburg Minisymposium on Pseudorandom Number
Generation and Quasi-Monte Carlo Methods, Salzburg, Nov 18, 1994, volume
ACPC/TR 95-4 of Technical Report Series, pages 109-121. ACPC -
Austrian Center for Parallel Computation, University of Vienna, Austria,
1995.
Abstract and postscript file available.
- 4
- H. Leeb and S. Wegenkittl. Inversive and linear congruential pseudorandom number generators in selected empirical tests. ACM TOMACS, 1997. to appear.