## SERIAL TEST

The serial (or digit) test based on digit expansion uses non-overlapping

s-tuples of integers defined by thek-th to thek+l-1-th bit of every pseudorandom number. For a sample sizen, the test is defined as follows: compute a chi-square test fromnnon-overlapping integers-tuples by observing how often each of the possible possible tuples occurs; repeat thismtimes; from themcorresponding chi-square values compute the KS statistic (denoted byt2(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, , andm= 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`

fors=4, and (in this and the following pictures, the values of are truncated to to keep the graphics in scale; note that ):

(2)`fishm`

fors=4, and :

(3)`icg1`

fors=4, and :

## 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 .
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 ofTechnical 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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