Testing multivariate uniformity and its applications

Liang, Jia-Juan; Fang, Kai-Tai; Hickernell, Fred; Li, Runze

doi:10.1090/S0025-5718-00-01203-5

Testing multivariate uniformity and its applications
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by Jia-Juan Liang, Kai-Tai Fang, Fred J. Hickernell and Runze Li PDF

Math. Comp. 70 (2001), 337-355 Request permission

Abstract:

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube $[0,1]^d\ (d\ge 2).$ These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in $[0,1]^d$. Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in $[0,1]^d$, we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, $N(0,1)$, or the chi-squared distribution, $\chi ^2(2)$. A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

References

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Ralph B. D’Agostino and Michael A. Stephens (eds.), Goodness-of-fit techniques, Statistics: Textbooks and Monographs, vol. 68, Marcel Dekker, Inc., New York, 1986. MR 874534
H. B. Fang, K. T. Fang, and S. Kotz, The meta-elliptical distributions with given marginals, Tech. Report MATH-165, Hong Kong Baptist University, 1997.
Kai Tai Fang, Samuel Kotz, and Kai Wang Ng, Symmetric multivariate and related distributions, Monographs on Statistics and Applied Probability, vol. 36, Chapman and Hall, Ltd., London, 1990. MR 1071174, DOI 10.1007/978-1-4899-2937-2
K.-T. Fang and Y. Wang, Number-theoretic methods in statistics, Monographs on Statistics and Applied Probability, vol. 51, Chapman & Hall, London, 1994. MR 1284470, DOI 10.1007/978-1-4899-3095-8
A. K. Gupta and D. Song, $L_p$-norm spherical distribution, J. Statist. Plann. Inference 60 (1997), no. 2, 241–260. MR 1456629, DOI 10.1016/S0378-3758(96)00129-2
Fred J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp. 67 (1998), no. 221, 299–322. MR 1433265, DOI 10.1090/S0025-5718-98-00894-1
F. J. Hickernell, Lattice rules: How well do they measure up?, Random and Quasi-Random Point Sets (P. Hellekalek and G. Larcher, eds.), Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, 1998, pp. 109–166.
F. J. Hickernell, Goodness-of-fit statistics, discrepancies and robust designs, Statist. Probab. Lett. 44 (1999), 73–78.
Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192, DOI 10.1007/978-3-642-67829-5
Ana Justel, Daniel Peña, and Rubén Zamar, A multivariate Kolmogorov-Smirnov test of goodness of fit, Statist. Probab. Lett. 35 (1997), no. 3, 251–259. MR 1484961, DOI 10.1016/S0167-7152(97)00020-5
S. Kotz and J. P. Seeger, A new approach to dependence in multivariate distributions, Advances in probability distributions with given marginals (Rome, 1990) Math. Appl., vol. 67, Kluwer Acad. Publ., Dordrecht, 1991, pp. 113–127. MR 1215948
F. L. Miller Jr. and C. P. Quesenberry, Power studies of tests for uniformity II, Comm. Statist. Simulation Comput. B8(3) (1979), 271–290.
J. Neyman, “Smooth" test for goodness of fit, J. Amer. Statist. Assoc. 20 (1937), 149–199.
Harald Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
E. S. Pearson, The probabilty transformation for testing goodness of fit and combining independent tests of significance, Biometrika 30 (1939), 134–148.
C. P. Quesenberry and F. L. Miller Jr., Power studies of some tests for uniformity, J. Statist. Comput. Simulation 5 (1977), 169–192.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Robert J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595165, DOI 10.1002/9780470316481
I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1442955
Yoshihiro Tashiro, On methods for generating uniform random points on the surface of a sphere, Ann. Inst. Statist. Math. 29 (1977), no. 2, 295–300. MR 488620, DOI 10.1007/BF02532791
S. Tezuka, Uniform random numbers: Theory and practice, Kluwer Academic Publishers, Boston, 1995.
G. S. Watson, Goodness-of-fit tests on a circle. II, Biometrika 49 (1962), 57–63. MR 138179, DOI 10.1093/biomet/49.1-2.57
Xin Nian Yue and Chunsheng Ma, Multivariate $l_p$-norm symmetric distributions, Statist. Probab. Lett. 24 (1995), no. 4, 281–288. MR 1353885, DOI 10.1016/0167-7152(94)00185-B