Linear and quadratic sufficiency and commutativity
Sandra Saraiva Ferreira, Dário Ferreira, Célia Nunes · AIP conference proceedings · 2012
Given a mixed model let T be the orthogonal projection matrix on the range space spanned by the mean vector. If the model has variance-covariance matrix σ2V we use commutative Jordan algebras to show that Ty is both linear sufficient and linear complete and that Ty, y′V+y with V+ the Moore-Penrose inverse of V is quadratic sufficient whenever T and V commute.