Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations

On the existence of the Gauss-Markov estimators in linear mixed models

“…2, a multivariate mixed linear model for the random n × p matrix Y of the observations is presented in a coordinate-free approach and the equivalence between the condition in Sect. 1 and some classical results [8,20] is proved for this model. In Sect.…”

“…The linear model M(E, V) is defined [8] as the set of all K-valued random elements y such that E(y) ∈ E ⊆ K and cov(y) = V ∈ V, where E is a linear subspace (or at least a linear manifold) of K and V is a convex cone of symmetric and nonnegative definite (n.n.d.) mappings V : K → K.…”

“…Using the general linear model from the coordinate-free point of view and a result given by Drygas [8] a necessary and sufficient condition for the existence of the G. Beganu ( ) Department of Mathematics, Academy of Economic Studies, Bucharest, Romania e-mail: gabriela_beganu@yahoo.com GME of the expected mean is proved and it is shown its equivalence to two similar conditions for the GME to exist and to be equal to the ordinary least squares estimators (OLSE) of the mean: one of them refers to the orthogonality of the ranges corresponding to the design matrices of the model [14,16,17] and the other one is a Zyskind's condition [20]. Thus the existence condition of the GME of the fixed effects becomes more easily to be verified.…”

“…In order to simplify the verifying of the existence conditions for the BQUE of the covariance components we will prove that the necessary and sufficient condition given by Seely [19] for the BQUE of the covariance components to exist in the initial linear model is equivalent to the condition given by Drygas [8] for the GME of the mean to exist in a new linear model built with an already known procedure [9].…”

“…3, a new linear model is built for the random matrix U = (vec Y )(vec Y ) and it is used to estimate the covariance components of Y . The equivalence between the necessary and sufficient conditions for the GME of E(U ) [8] and the BQUE of the covariance components [19] to exist in two different models is shown, using the coordinate-free approach. This procedure represents a more convenient way to verify the existence of the BQUE of the covariance components and it is applied to a family of multivariate repeated measurements with random effects.…”