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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Testing multivariate uniformity and its applications


Authors: Jia-Juan Liang, Kai-Tai Fang, Fred J. Hickernell and Runze Li
Journal: Math. Comp. 70 (2001), 337-355
MSC (2000): Primary 65C05, 62H10, 65D30
Published electronically: February 17, 2000
MathSciNet review: 1680903
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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.


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Additional Information

Jia-Juan Liang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China, and Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China
Email: jjliang@hkbu.edu.hk

Kai-Tai Fang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China, and Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China
Email: ktfang@hkbu.edu.hk

Fred J. Hickernell
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
Email: fred@hkbu.edu.hk

Runze Li
Affiliation: Department of Statistics, University of North Carolina, Chapel Hill, NC, 27599-3260, United States of America
Email: lirz@email.unc.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-00-01203-5
PII: S 0025-5718(00)01203-5
Keywords: Goodness-of-fit, discrepancy, quasi-Monte Carlo methods, testing uniformity
Received by editor(s): August 14, 1998
Received by editor(s) in revised form: February 11, 1999
Published electronically: February 17, 2000
Additional Notes: This work was partially supported by a Hong Kong Research Grants Council grant RGC/97-98/47.
Article copyright: © Copyright 2000 American Mathematical Society