Biadditive models: Commutativity and optimum estimators

Bi-additive models, are given by the sum of a fixed effects term X beta and w independent random terms X1Z1,..., X(w)Z(w), the components of Z(1),...,Z(w) being independent and identically distributed (i.i.d.) with null mean values and variances sigma(1)(2),...,sigma(2)(w). Thus besides having an additive structure they have covariance matrix Sigma(w)(i=1) sigma M-2(i)i, with M-i = XiXit,i = 1,...,w, thus their name. When matrices M-1,....,M-w, commute the covariance matrix will be a linear combination Sigma(m)(j=1) gamma(j)Q(j) of known, pairwise orthogonal, orthogonal projection matrices and we obtain BQUE for the gamma(1),...,gamma(m) through an extension of the HSU theorem and, when these matrices also commute with M=XXt, we also derive BLUE for gamma. The case in which the Z(1),...,Z(w) are normal is singled out and we then also obtain BQUE for the sigma(2)(1),...,sigma(2)(w). The interest of these models is that the types of the distributions of the components of vectors Z(1),...,Z(w) may belong to a wide family. This enlarges the applications of mixed models which has been centered on the normal type.