Tests for Kronecker envelope models in multilinear principal components analysis

We develop likelihood methods for the Kronecker envelope model in the principal components analysis of matrix observations that have a multivariate normal distribution. Maximum likelihood estimates are derived and the associated likelihood ratio statistic for a test of this Knonecker envelope model is obtained. The asymptotic null distribution of the likelihood ratio statistic is derived as some nuisance parameters approach infinity, and a saddlepoint approximation for this limiting distribution is given. An alternative composite test for the Kronecker envelope model, which can be used when the sample size is too small to use the likelihood ratio test, is also given. Simulation results demonstrate the accuracy of our approximations