School of Mathematics and Physics,
North China Electric Power University,College of Applied Science,
Beijing University of Technology

The $\chi^{2}$ conditional test for
multivariate normality is suggested. The transformed sample
$\mathbf{Y}_{d}=R\mathbf{V}_{d}$ from a $d$-variate normal
distribution has a symmetric multivariate Pearson type II
distribution, the result that $R^{2}$ has a beta distribution is
proved, the asymptotic Chi squared distribution of the statistic
$\chi^{2}$ based on beta distribution and sphere uniform
distribution is obtained. The Monte Carlo power study for
multivariate normality suggests that our test is a powerful
competitor to existing tests. The goodness-of-fit for multivariate
normality of iris data is analyzed.