Coherence and Classical Tests in the Multivariate Normal Model

In this chapter, we establish many basic results concerning inference and hypothesis testing in the proper, complex, multivariate normal distribution. We consider in particular second-order measurement models in which the unknown covariance matrix belongs to a cone. This is often the case in signal processing and machine learning. Two important results concerning maximum likelihood (ML) estimators and likelihood ratios computed from ML estimators are reviewed. We then proceed to examine several classical hypothesis tests about the covariance matrix of measurements in multivariate normal (MVN) models. These are the sphericity test that tests whether or not the covariance matrix is a scaled identity matrix with unknown scale parameter; the Hadamard test that tests whether or not the variables in a MVN model are independent, thus having a diagonal covariance matrix with unknown diagonal elements; and the homogeneity test that tests whether or not the covariance matrices of independent vector-valued MVN models are equal. The chapter concludes with a discussion of the expected likelihood principle for cross-validating a covariance model.