Biadditive models: Commutativity and optimum estimators

Abstract

Bi-additive models, are given by the sum of a fixed effects term X β and w independent random terms X 1 Z 1 , … , 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 σ 1 2 , … , σ w 2 . Thus besides having an additive structure they have covariance matrix ∑ i = 1 w σ i 2 M i , with M i = X i X i t , i = 1 , … , w , thus their name. When matrices M 1 , … , M w , commute the covariance matrix will be a linear combination ∑ j = 1 m γ j Q j of known, pairwise orthogonal, orthogonal projection matrices and we obtain BQUE for the γ 1 , … , γ m through an extension of the HSU theorem and, when these matrices also commute with M = X X t , we also derive BLUE for γ . The case in which the Z 1 , … , Z w are normal is singled out and we then also obtain BQUE for the σ 1 2 , … , σ w 2 . 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.