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

DOI:
https://doi.org/10.1090/S0025-5718-00-01203-5

Published electronically:
February 17, 2000

MathSciNet review:
1680903

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in , 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, , or the chi-squared distribution, . 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:
https://doi.org/10.1090/S0025-5718-00-01203-5

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