{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:25:49Z","timestamp":1776831949878,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,2,17]],"date-time":"2001-02-17T00:00:00Z","timestamp":982368000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n \n ]<\/mml:mo>\n d<\/mml:mi>\n <\/mml:msup>\n \u00a0<\/mml:mtext>\n (<\/mml:mo>\n d<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n [0,1]^d\\ (d\\ge 2).<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n \n ]<\/mml:mo>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n [0,1]^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n \n ]<\/mml:mo>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n [0,1]^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , 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 \n \n \n \n N<\/mml:mi>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n N(0,1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , or the chi-squared distribution,\n \n \n \n \n \n \n \u03c7\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \\chi ^2(2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.\n <\/p>","DOI":"10.1090\/s0025-5718-00-01203-5","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T17:02:26Z","timestamp":1121101346000},"page":"337-355","source":"Crossref","is-referenced-by-count":38,"title":["Testing multivariate uniformity and its applications"],"prefix":"10.1090","volume":"70","author":[{"given":"Jia-Juan","family":"Liang","sequence":"first","affiliation":[]},{"given":"Kai-Tai","family":"Fang","sequence":"additional","affiliation":[]},{"given":"Fred","family":"Hickernell","sequence":"additional","affiliation":[]},{"given":"Runze","family":"Li","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,2,17]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"611","DOI":"10.2307\/1968946","article-title":"On Frobeniusean algebras. 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