Statistical properties of 𝑏-adic diaphonies

Seri, Raffaello

doi:10.1090/mcom/3148

Statistical properties of $b$-adic diaphonies
HTML articles powered by AMS MathViewer

by Raffaello Seri PDF

Math. Comp. 86 (2017), 799-828 Request permission

Abstract:

The aim of this paper is to derive the asymptotic statistical properties of a class of discrepancies on the unit hypercube called $b$-adic diaphonies. They have been introduced to evaluate the equidistribution of quasi-Monte Carlo sequences on the unit hypercube. We consider their properties when applied to a sample of independent and uniformly distributed random points. We show that the limiting distribution of the statistic is an infinite weighted sum of chi-squared random variables, whose weights can be explicitly characterized and computed. We also describe the rate of convergence of the finite-sample distribution to the asymptotic one and show that this is much faster than in the classical Berry-Esséen bound. Then, we consider in detail the approximation of the asymptotic distribution through two truncations of the original infinite weighted sum, and we provide explicit and tight bounds for the truncation error. Numerical results illustrate the findings of the paper, and an empirical example shows the relevance of the results in applications.

References

Clemens Amstler and Peter Zinterhof, Uniform distribution, discrepancy, and reproducing kernel Hilbert spaces, J. Complexity 17 (2001), no. 3, 497–515. MR 1851057, DOI 10.1006/jcom.2001.0580
V. Bentkus and F. Götze, Optimal bounds in non-Gaussian limit theorems for $U$-statistics, Ann. Probab. 27 (1999), no. 1, 454–521. MR 1681161, DOI 10.1214/aop/1022677269
Christine Choirat and Raffaello Seri, The asymptotic distribution of quadratic discrepancies, Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin, 2006, pp. 61–76. MR 2208702, DOI 10.1007/3-540-31186-6_{5}
Christine Choirat and Raffaello Seri, Statistical properties of generalized discrepancies, Math. Comp. 77 (2008), no. 261, 421–446. MR 2353960, DOI 10.1090/S0025-5718-07-01839-X
Christine Choirat and Raffaello Seri, Computational aspects of Cui-Freeden statistics for equidistribution on the sphere, Math. Comp. 82 (2013), no. 284, 2137–2156. MR 3073194, DOI 10.1090/S0025-5718-2013-02698-1
Ronald Christensen, Testing Fisher, Neyman, Pearson, and Bayes, Amer. Statist. 59 (2005), no. 2, 121–126. MR 2133558, DOI 10.1198/000313005X20871
Ligia L. Cristea and Friedrich Pillichshammer, A lower bound for the $b$-adic diaphony, Rend. Mat. Appl. (7) 27 (2007), no. 2, 147–153. MR 2361027
R. B. Davies, Numerical inversion of a characteristic function, Biometrika 60 (1973), 415–417. MR 321152, DOI 10.1093/biomet/60.2.415
R. B. Davies, Statistical algorithms: Algorithm AS 155: The distribution of a linear combination of $\chi ^2$ random variables, Applied Statistics 29 (1980), no. 3, 323–333.
Herold Dehling, Complete convergence of triangular arrays and the law of the iterated logarithm for $U$-statistics, Statist. Probab. Lett. 7 (1989), no. 4, 319–321. MR 980708, DOI 10.1016/0167-7152(89)90115-6
Josef Dick and Friedrich Pillichshammer, Diaphony, discrepancy, spectral test and worst-case error, Math. Comput. Simulation 70 (2005), no. 3, 159–171. MR 2176902, DOI 10.1016/j.matcom.2005.06.004
Josef Dick and Friedrich Pillichshammer, Dyadic diaphony of digital nets over $\Bbb Z_2$, Monatsh. Math. 145 (2005), no. 4, 285–299. MR 2162347, DOI 10.1007/s00605-004-0287-7
Josef Dick and Friedrich Pillichshammer, Digital nets and sequences, Cambridge University Press, Cambridge, 2010. Discrepancy theory and quasi-Monte Carlo integration. MR 2683394, DOI 10.1017/CBO9780511761188
Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics, vol. 1651, Springer-Verlag, Berlin, 1997. MR 1470456, DOI 10.1007/BFb0093404
Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1–125. MR 14626, DOI 10.1007/BF02392223
Chelo Ferreira and José L. López, Asymptotic expansions of the Hurwitz-Lerch zeta function, J. Math. Anal. Appl. 298 (2004), no. 1, 210–224. MR 2086542, DOI 10.1016/j.jmaa.2004.05.040
Evarist Giné M., Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms, Ann. Statist. 3 (1975), no. 6, 1243–1266. MR 388663
S. J. Gould, Glow, big glowworm, Natural History 12 (1986), 10–16 (English).
S. J. Gould, Bully for Brontosaurus: Reflections in Natural History, W. W. Norton, 1991.
Gavin G. Gregory, Large sample theory for $U$-statistics and tests of fit, Ann. Statist. 5 (1977), no. 1, 110–123. MR 433669
Julia Greslehner, The $b$-adic diaphony of digital sequences, Unif. Distrib. Theory 5 (2010), no. 2, 87–112. MR 2781411
Julia Greslehner, The $b$-adic diaphony of digital $(\mathbf T,s)$-sequences, Unif. Distrib. Theory 6 (2011), no. 2, 1–12. MR 2904034
V. Grozdanov, The generalized $b$-adic diaphony of the net of Zaremba-Halton over finite abelian groups, C. R. Acad. Bulgare Sci. 57 (2004), no. 6, 43–48. MR 2075788
V. Grozdanov, The weighted $b$-adic diaphony, C. R. Acad. Bulgare Sci. 58 (2005), no. 5, 493–498. MR 2154330
Vassil Grozdanov, The weighted $b$-adic diaphony, J. Complexity 22 (2006), no. 4, 490–513. MR 2246893, DOI 10.1016/j.jco.2005.11.008
V. S. Grozdanov and E. G. Miteva, On the dyadic diaphony of the Zaremba sequence, C. R. Acad. Bulgare Sci. 53 (2000), no. 4, 5–8. MR 1780010
V. Grozdanov, E. Nikolova, and S. Stoilova, Generalized $b$-adic diaphony, C. R. Acad. Bulgare Sci. 56 (2003), no. 4, 23–30. MR 1980679
V. S. Grozdanov and S. S. Stoilova, On the theory of $b$-adic diaphony, C. R. Acad. Bulgare Sci. 54 (2001), no. 3, 31–34. MR 1829550
V. S. Grozdanov and S. S. Stoilova, The $b$-adic diaphony, Rend. Mat. Appl. (7) 22 (2002), 203–221 (2003) (English, with English and Italian summaries). MR 2041234
V. S. Grozdanov and S. S. Stoilova, On the irregularities of distribution, C. R. Acad. Bulgare Sci. 55 (2002), no. 3, 15–18. MR 1882673
V. S. Grozdanov and S. S. Stoilova, On the $b$-adic diaphony of the Roth net and generalized Zaremba net, Math. Balkanica (N.S.) 17 (2003), no. 1-2, 103–112. MR 2096244
V. Grozdanov and S. Stoilova, The general diaphony, C. R. Acad. Bulgare Sci. 57 (2004), no. 1, 13–18. MR 2032921
Peter Hellekalek and Hannes Leeb, Dyadic diaphony, Acta Arith. 80 (1997), no. 2, 187–196. MR 1450924, DOI 10.4064/aa-80-2-187-196
J. P. Imhof, Computing the distribution of quadratic forms in normal variables, Biometrika 48 (1961), 419–426. MR 137199, DOI 10.1093/biomet/48.3-4.419
Peter Imkeller, On exact tails for limiting distributions of $U$-statistics in the second Gaussian chaos, Chaos expansions, multiple Wiener-Itô integrals and their applications (Guanajuato, 1992) Probab. Stochastics Ser., CRC, Boca Raton, FL, 1994, pp. 179–204. MR 1278044
Peter Kritzer and Friedrich Pillichshammer, The weighted dyadic diaphony of digital sequences, Monte Carlo and quasi-Monte Carlo methods 2006, Springer, Berlin, 2008, pp. 549–560. MR 2479245, DOI 10.1007/978-3-540-74496-2_{3}2
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
Hannes Leeb, Asymptotic properties of the spectral test, diaphony, and related quantities, Math. Comp. 71 (2002), no. 237, 297–309. MR 1863001, DOI 10.1090/S0025-5718-01-01356-4
E. L. Lehmann, The Fisher, Neyman-Pearson theories of testing hypotheses: one theory or two?, J. Amer. Statist. Assoc. 88 (1993), no. 424, 1242–1249. MR 1245356
Ivan Lirkov and Stanislava Stoilova, The $b$-adic diaphony as a tool to study pseudo-randomness of nets, Numerical methods and applications, Lecture Notes in Comput. Sci., vol. 6046, Springer, Heidelberg, 2011, pp. 68–76. MR 2794677, DOI 10.1007/978-3-642-18466-6_{7}
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
Friedrich Pillichshammer, Dyadic diaphony of digital sequences, J. Théor. Nombres Bordeaux 19 (2007), no. 2, 501–521 (English, with English and French summaries). MR 2394899
S. Pinker, The Better Angels of Our Nature: The Decline of Violence in History and its Causes, Penguin Books Limited, 2011.
S. O. Rice, Distribution of quadratic forms in normal random variables—evaluation by numerical integration, SIAM J. Sci. Statist. Comput. 1 (1980), no. 4, 438–448. MR 610756, DOI 10.1137/0901032
V. Ristovska, Irregularity of the distribution of the generalized van der Corput sequence, C. R. Acad. Bulgare Sci. 58 (2005), no. 5, 487–492. MR 2154329
H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
Vjačeslav V. Sazonov, Normal approximation—some recent advances, Lecture Notes in Mathematics, vol. 879, Springer-Verlag, Berlin-New York, 1981. MR 643968
Robert J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595165
R. Seri, A tight bound on the distance between a noncentral chi square and a normal distribution, IEEE Communications Letters 19 (2015), no. 11, 1877–1880.
J. Sheil and I. O’Muircheartaigh, Statistical algorithms: Algorithm AS 106: The distribution of non-negative quadratic forms in normal variables, Applied Statistics 26 (1977), no. 1, 92–98.
Galen R. Shorack, Probability for statisticians, Springer Texts in Statistics, Springer-Verlag, New York, 2000. MR 1762415
S. S. Stoilova, The $b$-adic diaphony of an arbitrary $(t,m,s)$-net, Tech. Report E5-2004-107, Joint Institute for Nuclear Research, Dubna (Russia), 2004.
S. S. Stoilova, On the $b$-adic diaphony of the generalized Zaremba net, Tech. Report E5-2005-96, Joint Institute for Nuclear Research, Dubna (Russia), 2005.
A. W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3, Cambridge University Press, Cambridge, 1998. MR 1652247, DOI 10.1017/CBO9780511802256
P. Zinterhof, Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 185 (1976), no. 1-3, 121–132 (German). MR 0501760