Goodness-of-fit analysis for uniformity in a bounded solid region

The test statistic χ² for uniformity in a bounded solid region D₁ is suggested. The sufficient and necessary condition for the uniform distribution in D₁ is proved, the asymptotic chi squared distribution of χ² is obtained, the consistency of the goodness-of-fit tests is proved. The simulation of the convergence rate of the empirical distribution function and the test power is made, the simulation results show that the multivariate goodness-of-fit tests for sample size n ≥ 10 is effective.