  In this article, we propose a new class of consistent tests for $p$-variate
normality. These tests are based on the characterization of the standard
multivariate normal distribution, that the Hessian of the corresponding
cumulant generating function is identical to the $p\times p$ identity matrix
and the idea of decomposing the information from the joint distribution into
the dependence copula and all marginal distributions. Under the null hypothesis
of multivariate normality, our proposed test statistic is independent of the
unknown mean vector and covariance matrix so that the distribution-free
critical value of the test can be obtained by Monte Carlo simulation. We also
derive the asymptotic null distribution of proposed test statistic and
establish the consistency of the test against different fixed alternatives.
Last but not least, a comprehensive and extensive Monte Carlo study also
illustrates that our test is a superb yet computationally convenient competitor
to many well-known existing test statistics.
