Biadditive models: Commutativity and optimum estimators

Bi-additive models, are given by the sum of a fixed effects term Xβ and w independent random terms X1Z1,…, XwZw, the components of Z1,…,Zw being independent and identically distributed (i.i.d.) with null mean values and variances σ12,…,σw2. Thus besides having an additive structure they have covariance matrix ∑i=1wσi2Mi, with Mi=XiXit,i=1,…,w, thus their name. When matrices M1,…,Mw, commute the covariance matrix will be a linear combination ∑j=1mγjQj of known, pairwise orthogonal, orthogonal projection matrices and we obtain BQUE for the γ1,…,γm through an extension of the HSU theorem and, when these matrices also commute with M=XXt, we also derive BLUE for γ. The case in which the Z1,…,Zw are normal is singled out and we then also obtain BQUE for the σ12,…,σw2. The interest of these models is that the types of the distributions of the components of vectors Z1,…,Zw may belong to a wide family. This enlarges the applications of mixed models which has been centered on the normal type.