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 greater than or equal to 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)., 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.