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 tex2html_wrap_inline69 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 tex2html_wrap_inline77 the excellent linear generator tex2html_wrap_inline79 and by the inversive congruential generator tex2html_wrap_inline81 for s=4, tex2html_wrap_inline85, tex2html_wrap_inline87 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 tex2html_wrap_inline91 with respect to their lattice structure in dimensions tex2html_wrap_inline93.

(1) ansi for s=4, tex2html_wrap_inline85 and tex2html_wrap_inline99 (in this and the following pictures, the values of tex2html_wrap_inline101 are truncated to tex2html_wrap_inline103 to keep the graphics in scale; note that tex2html_wrap_inline105 ):
tabular23

(2) fishm for s=4, tex2html_wrap_inline85 and tex2html_wrap_inline99:
tabular29

(3) icg1 for s=4, tex2html_wrap_inline85 and tex2html_wrap_inline99:
tabular35

References

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 tex2html_wrap_inline142. 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.


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Karl Entacher
Mon Jun 9 15:54:14 MET DST 1997