Testing multivariate uniformity and its applications

Abstract

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube [ 0 , 1 ] d   ( d ≥ 2 ) . [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 [0,1]^d . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in [ 0 , 1 ] d [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 ) N(0,1) , or the chi-squared distribution, χ 2 ( 2 ) \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.