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 ):

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

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


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.

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.

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.

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