Estimation of parameters under a generalized growth curve model

This paper concerns multi-level multivariate data. Such data can be presented in the form of a multi-index matrix (tensor) Y. First the third-order normally distributed tensor of observations, Y∈Rn×p×q, is discussed with the mean structured in the form of a generalized growth curve model, [[X;A,B,C]], with multiplication in all three directions of the third-order tensor X of unknown parameters by the known matrices A, B and C. The paper is focused on the estimation of an unknown tensor X of direct effects and a separable and doubly separable variance-covariance matrix. Since the resulting estimators of unknown parameters cannot be presented in an explicit form, the estimates are obtained approximately. The uniqueness of the so-called 'flip-flop' algorithm is also discussed, and the use of the algorithm is illustrated on a real data example. Finally, possible extensions of the third-order generalized growth curve model to more levels are considered.