Multivariate linear normal models with special references to the growth curve model

Abstract: The Growth Curve Model (GMAN0VA) introduced by Potthoff & Roy (1964) and two extensions are considered. The first extension treats a Growth Curve Model with concomitant variables whereas the second is applicable when different growth curves or when linear restrictions on the parameter space, describing the mean structure in the Growth Curve Model, exist. Likelihood equations for the Growth Curve Model as well as for the proposed extensions are solved and it is established that the solutions are maximum likelihood estimators. For each solution necessary and sufficient conditions for uniqueness are given. When the estimators or linear functions of these are unique, moments are derived. Among others, the first five moments of the estimator describing the mean structure in the Growth Curve Model are given as well as the first and second moments for the estimator of the dispersion matrix. Furthermore, asymptotic equivalent expressions for the maximum likelihood estimators which are normally distributed are presented. In order to obtain the above mentioned results quite a lot of matrix algebra and moment relations are needed. Lemmas which include original work on vector spaces, linear equations, Kronecker products, the commutation matrix, matrix derivatives, moments for matrix normally distributed variables, moments of quadratic forms and moments for the inverted Wishart distribution are given.