Testing multivariate uniformity and its applications

Some new Statistics are proposed to test the uniformity of random samples i n 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 met hods for measuring the discrepancy of points in [0, 1](d). Under the null h ypothesis 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 st andard normal distribution, N(0, 1), or the chi-squared distribution, chi ( 2)(2). A power study is performed, and possible applications of the new sta tistics to testing general multivariate goodness-of-fit problems are discus sed.