Testing multivariate uniformity and its applications |
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| Authors: | Jia-Juan Liang Kai-Tai Fang Fred J Hickernell Runze Li |
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| Institution: | Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China, and Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China ; Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China, and Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China ; Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China ; Department of Statistics, University of North Carolina, Chapel Hill, NC, 27599-3260, United States of America |
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| Abstract: | Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube ![$0,1]^d (d\ge 2).$](http://www.ams.org/mcom/2001-70-233/S0025-5718-00-01203-5/gif-abstract0/img1.gif) These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in ![$0,1]^d$](http://www.ams.org/mcom/2001-70-233/S0025-5718-00-01203-5/gif-abstract0/img2.gif) . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in ![$0,1]^d$](http://www.ams.org/mcom/2001-70-233/S0025-5718-00-01203-5/gif-abstract0/img3.gif) , 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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| Keywords: | Goodness-of-fit discrepancy quasi-Monte Carlo methods testing uniformity |
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