Biadditive models: Commutativity and optimum estimators

Bi-additive models, are given by the sum of a fixed effects term (Formula presented.) and w independent random terms (Formula presented.) (Formula presented.) the components of (Formula presented.) being independent and identically distributed (i.i.d.) with null mean values and variances (Formula presented.) Thus besides having an additive structure they have covariance matrix (Formula presented.) with (Formula presented.) thus their name. When matrices (Formula presented.) commute the covariance matrix will be a linear combination (Formula presented.) of known, pairwise orthogonal, orthogonal projection matrices and we obtain BQUE for the (Formula presented.) through an extension of the HSU theorem and, when these matrices also commute with (Formula presented.) we also derive BLUE for (Formula presented.) The case in which the (Formula presented.) are normal is singled out and we then also obtain BQUE for the (Formula presented.) The interest of these models is that the types of the distributions of the components of vectors (Formula presented.) may belong to a wide family. This enlarges the applications of mixed models which has been centered on the normal type.