Estimation of parameters in the growth curve model via an outer product least squares approach for covariance

In this paper, we propose a framework of outer product least squares for covariance (COPLS) to directly estimate covariance in the growth curve model based on an analogy, between the outer product of a data vector and covariance of a random vector, and the ordinary least squares technique. The COPLS estimator of covariance has an explicit expression and is shown to have the following properties: (1) following a linear transformation of two independent Wishart distribution for a normal error matrix; (2) having asymptotic normality for a nonnormal error matrix; and (3) having unbiasedness and invariance under a linear transformation group. And, a corresponding two-stage generalized least squares (GLS) estimator for the regression coefficient matrix in the model is obtained and its asymptotic normality is investigated. Simulation studies confirm that the COPLS estimator and the two-stage GLS estimator of the regression coefficient matrix are satisfying competitors with some evident merits to the existing maximum likelihood estimator in finite samples.