Testing multivariate uniformity and its applications

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.