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Close-Point Spatial Tests and Their Application to Random Number Generators

Author

Listed:
  • Pierre L'Écuyer

    (Département ďInformatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada)

  • Jean-François Cordeau

    (Département ďInformatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada)

  • Richard Simard

    (Département ďInformatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada)

Abstract

We study statistical tests of uniformity based on the L p -distances between the m nearest pairs of points, for n points generated uniformly over the k -dimensional unit hypercube or unit torus. The number of distinct pairs at distance no more than t , for t (ge) 0, is a stochastic process whose initial part, after an appropriate transformation and as n (rightarrow) (infinity), is asymptotically a Poisson process with unit rate. Convergence to this asymptotic is slow in the hypercube as soon as k exceeds 2 or 3, due to edge effects, but is reasonably fast in the torus. We look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions, discuss computational issues, and apply the tests to random number generators. Linear congruential generators fail decisively certain variants of the tests as soon as n approaches the square root of the period length.

Suggested Citation

  • Pierre L'Écuyer & Jean-François Cordeau & Richard Simard, 2000. "Close-Point Spatial Tests and Their Application to Random Number Generators," Operations Research, INFORMS, vol. 48(2), pages 308-317, April.
    • Handle: RePEc:inm:oropre:v:48:y:2000:i:2:p:308-317
    DOI: 10.1287/opre.48.2.308.12385
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    Cited by:

    1. Haramoto, Hiroshi & Matsumoto, Makoto, 2019. "Checking the quality of approximation of p-values in statistical tests for random number generators by using a three-level test," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 161(C), pages 66-75.
    2. Harase, Shin, 2019. "Conversion of Mersenne Twister to double-precision floating-point numbers," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 161(C), pages 76-83.
    3. Grace, Adam W. & Wood, Ian A., 2012. "Approximating the tail of the Anderson–Darling distribution," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4301-4311.

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