Maximum-likelihood estimation of the parameters of a multivariate normal distribution

AbstractThis paper provides an exposition of alternative approaches for obtaining maximum- likelihood estimators (MLE) for the parameters of a multivariate normal distribution under different assumptions about the parameters. A central focus is on two general techniques, namely, matrix differentiation and matrix transformations. These are systematically applied to derive the MLE of the means under a rank constraint and of the covariances when there are missing observations. Derivations using induction and inequalities are also included to illustrate alternative methods. Other examples, such as a connection with an econometric model, are included. Although the paper is primarily expository, some of the proofs are new