Advertisement

The Asymptotic Distribution of Quadratic Discrepancies

  • Christine Choirat
  • Raffaello Seri

Summary

In Numerical Analysis, several discrepancies have been introduced to test that a sample of n points in the unit hypercube [0, 1]d comes from a uniform distribution. An outstanding example is given by Hickernell’s generalized \(\mathcal{L}^P \)-discrepancies, that constitute a generalization of the Kolmogorov-Smirnov and the Cramér-von Mises statistics. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity and goodness of fit tests. In this paper, after having recalled some necessary asymptotic results derived in companion papers, we show that the case of \(\mathcal{L}^2 \)-discrepancies is more convenient to handle and we provide a new computational approximation of their asymptotic distribution. As an illustration, we show that our algorithm is able to recover the tabulated asymptotic distribution of the Cramér-von Mises statistic. The results so obtained are very general and can be applied with minor modifications to other discrepancies, such as the diaphony, the weighted spectral test, the Fourier discrepancy and the class of chi-square tests.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T.W. Anderson and D.A. Darling. Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Ann. Math. Statistics, 23:193–212, 1952.MathSciNetGoogle Scholar
  2. 2.
    R.H. Brown. The distribution function of positive definite quadratic forms in normal random variables. SIAM J. Sci. Statist. Comput., 7(2):689–695, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    C. Choirat and R. Seri. Statistical properties of generalized discrepancies. Working paper, 2004.Google Scholar
  4. 4.
    C. Choirat and R. Seri. Statistical properties of quadratic discrepancies. Working paper, 2004.Google Scholar
  5. 5.
    R.R. Coveyou. Review MR0351035 of MathSciNet, 1975.Google Scholar
  6. 6.
    S. Csörgő and J.J. Faraway. The exact and asymptotic distributions of Cramérvon Mises statistics. J. Roy. Statist. Soc. Ser. B, 58(1):221–234, 1996.MathSciNetGoogle Scholar
  7. 7.
    R.B. Davies. Numerical inversion of a characteristic function. Biometrika, 60:415–417, 1973.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    R.B. Davies. Statistical algorithms: Algorithm AS 155: The distribution of a linear combination of ϰ2 random variables. Applied Statistics, 29(3):323–333, 1980.zbMATHGoogle Scholar
  9. 9.
    K. Frank and S. Heinrich. Computing discrepancies of Smolyak quadrature rules. J. Complexity, 12(4):287–314, 1996. Special issue for the Foundations of Computational Mathematics Conference (Rio de Janeiro, 1997).CrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Gil-Pelaez. Note on the inversion theorem. Biometrika, 38:481–482, 1951.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    P.J. Grabner, P. Liardet, and R.F. Tichy. Average case analysis of numerical integration. In Advances in Multivariate Approximation (Witten-Bommerholz, 1998), volume 107 of Math. Res., pages 185–200. Wiley-VCH, Berlin, 1999.Google Scholar
  12. 12.
    P.J. Grabner, O. Strauch, and R.F. Tichy. Lp-discrepancy and statistical independence of sequences. Czechoslovak Math. J., 49(124)(1):97–110, 1999.CrossRefMathSciNetGoogle Scholar
  13. 13.
    V.S. Grozdanov and S.S. Stoilova. The b-adic diaphony. Rend. Mat. Appl. (7), 22:203–221 (2003), 2002.MathSciNetGoogle Scholar
  14. 14.
    P. Hellekalek and H. Niederreiter. The weighted spectral test: diaphony. ACM Trans. Model. Comput. Simul., 8(1):43–60, 1998.CrossRefGoogle Scholar
  15. 15.
    P. Hellekalek. Dyadic diaphony. Acta Arith., 80(2):187–196, 1997.zbMATHMathSciNetGoogle Scholar
  16. 16.
    P. Hellekalek. On correlation analysis of pseudorandom numbers. In Monte Carlo and Quasi-Monte Carlo Methods 1996 (Salzburg), volume 127 of Lecture Notes in Statist., pages 251–265. Springer, New York, 1998.Google Scholar
  17. 17.
    P. Hellekalek. On the assessment of random and quasi-random point sets. In Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statist., pages 49–108. Springer, New York, 1998.Google Scholar
  18. 18.
    F.J. Hickernell. Erratum: “Quadrature error bounds with applications to lattice rules” [SIAM J. Numer. Anal. 33 (1996), no. 5, 1995–2016;]. SIAM J. Numer. Anal., 34(2):853–866, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    F.J. Hickernell. Quadrature error bounds with applications to lattice rules. SIAM J. Numer. Anal., 33(5):1995–2016, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    F.J. Hickernell. A generalized discrepancy and quadrature error bound. Math. Comp., 67(221):299–322, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    F.J. Hickernell. Lattice rules: how well do they measure up? In Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statist., pages 109–166. Springer, New York, 1998.Google Scholar
  22. 22.
    F.J. Hickernell. Goodness-of-fit statistics, discrepancies and robust designs. Statist. Probab. Lett., 44(1):73–78, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    F.J. Hickernell. The mean square discrepancy of randomized nets. ACM Trans. Model. Comput. Simul., 6(4):274–296, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    F.J. Hickernell. What affects the accuracy of quasi-Monte Carlo quadrature? In Monte Carlo and Quasi-Monte Carlo Methods 1998 (Claremont, CA), pages 16–55. Springer, Berlin, 2000.Google Scholar
  25. 25.
    J. Hoogland, F. James, and R. Kleiss. Quasi-Monte Carlo, discrepancies and error estimates. In Monte Carlo and Quasi-Monte Carlo Methods 1996 (Salzburg), volume 127 of Lecture Notes in Statist., pages 266–276. Springer, New York, 1998.Google Scholar
  26. 26.
    J. Hoogland and R. Kleiss. Discrepancy-based error estimates for quasi-monte carlo. I: General formalism. Comput. Phys. Comm., 98(1–2):111–127, 1996.CrossRefGoogle Scholar
  27. 27.
    J. Hoogland and R. Kleiss. Discrepancy-based error estimates for quasi-monte carlo. II: Results in one dimension. Comput. Phys. Comm., 98(1–2):128–136, 1996.CrossRefGoogle Scholar
  28. 28.
    J. Hoogland and R. Kleiss. Discrepancy-based error estimates for quasi-monte carlo. III: Error distribution and central limits. Comput. Phys. Comm., 101(1–2):21–30, 1997.CrossRefMathSciNetGoogle Scholar
  29. 29.
    J.P. Imhof. Computing the distribution of quadratic forms in normal variables. Biometrika, 48:419–426, 1961.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    F. James, J. Hoogland, and R. Kleiss. Multidimensional sampling for simulation and integration: Measures, discrepancies and quasi-random numbers. Comput. Phys. Comm., 99(2–3):180–220, 1997.CrossRefGoogle Scholar
  31. 31.
    V. Koltchinskii and E. Giné. Random matrix approximation of spectra of integral operators. Bernoulli, 6(1):113–167, 2000.CrossRefMathSciNetGoogle Scholar
  32. 32.
    P. L’Ecuyer and P. Hellekalek. Random number generators: selection criteria and testing. In Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statist., pages 223–265. Springer, New York, 1998.Google Scholar
  33. 33.
    H. Leeb. Asymptotic properties of the spectral test, diaphony, and related quantities. Math. Comp., 71(237):297–309, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    V.F. Lev. On two versions of L2-discrepancy and geometrical interpretation of diaphony. Acta Math. Hungar., 69(4):281–300, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    J.-J. Liang, K.-T. Fang, F.J. Hickernell, and R. Li. Testing multivariate uniformity and its applications. Math. Comp., 70(233):337–355, 2001.CrossRefMathSciNetGoogle Scholar
  36. 36.
    W.J. Morokoff and R.E. Caflisch. Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput., 15(6):1251–1279, 1994.CrossRefMathSciNetGoogle Scholar
  37. 37.
    G. Pagès and Y.-J. Xiao. Sequences with low discrepancy and pseudo-random numbers: theoretical results and numerical tests. J. Statist. Comput. Simulation, 56(2):163–188, 1997.MathSciNetGoogle Scholar
  38. 38.
    S.H. Paskov. Average case complexity of multivariate integration for smooth functions. J. Complexity, 9(2):291–312, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2004. ISBN 3-900051-00-3.Google Scholar
  40. 40.
    S.O. Rice. Distribution of quadratic forms in normal random variables-evaluation by numerical integration. SIAM J. Sci. Statist. Comput., 1(4):438–448, 1980.CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    J. Sheil and I. O’Muircheartaigh. Statistical algorithms: Algorithm AS 106: The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26(1):92–98, 1977.Google Scholar
  42. 42.
    O. Strauch. L2 discrepancy. Math. Slovaca, 44(5):601–632, 1994. Number theory (Račkova dolina, 1993).zbMATHMathSciNetGoogle Scholar
  43. 43.
    A. van Hameren, R. Kleiss, and J. Hoogland. Gaussian limits for discrepancies. I. Asymptotic results. Comput. Phys. Comm., 107(1–3):1–20, 1997.MathSciNetGoogle Scholar
  44. 44.
    T.T. Warnock. Computational investigations of low-discrepancy point sets. In Applications of Number Theory to Numerical Analysis (Proc. Sympos., Univ. Montreal, Montreal, Que., 1971), pages 319–343. Academic Press, New York, 1972.Google Scholar
  45. 45.
    G.S. Watson. Goodness-of-fit tests on a circle. Biometrika, 48:109–114, 1961.CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    G.S. Watson. Another test for the uniformity of a circular distribution. Biometrika, 54:675–677, 1967.CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    P. Zinterhof. Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II, 185(1–3):121–132, 1976.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christine Choirat
    • 1
  • Raffaello Seri
    • 1
  1. 1.Dipartimento di EconomiaUniversità degli Studi dell’InsubriaVareseItaly

Personalised recommendations