Testing multivariate normality using angular skewness and kurtosis processes

Music City Center 

Given a sample X_1, ... , X_n from a common distribution P in R^d, d ≥ 2, we develop a method to test multivariate normality based on two random processes S_n and K_n indexed by the unit sphere S^{d-1}, which stand respectively for the skewness and kurtosis of linear combinations. We show that the limit processes are Gaussian and can be represented as random finite linear combinations of the spherical harmonics. We consider test statistics based on the supremums of these processes and numerically obtain their limit distributions. We also show that the Bayesian bootstrap can consistently estimate the cutoffs. We obtain the limiting power of the test under contiguous alternative hypotheses. Through an extensive simulation study, we show that our proposed method performs well for moderate and large sample sizes.

multivariate normality test

skewness and kurtosis

stochastic processes

spherical harmonics

emprical processes

Bayesian bootstrap 

IMS