Goodness-of-fit Analysis for Uniformity in a Unit Sphere

In order to test the uniformity in a unit sphere the χ² test statistic for uniformity in a unit sphere is suggested. The sufficient and necessary conditio the asymptotic Chi squared distribution of the χ² empirical distribution function and the test power for the uniform distribution in a unit sphere is proved, and is obtained. The simulation of the convergence rate of the is made. The simulation results show that the multivariate goodness-of-fit tests for sample size n≥10 is effective.