Linear Transformations Preserving Best Linear Unbiased Estimators in a General Gauss-Markoff Model

Best linear unbiased prediction for linear combinations in general mixed linear models

“…Furthermore, Ibarrola and Pérez-Palomares [17,18] applied linear sufficiency and linear completeness to a continuous time linear model and corresponding characterizations were offered. Baksalary and Kala [2] proved that Fy is linearly sufficient if and only if…”

“…The form of the model is a generalization of that of Fay and Herriot [11] in some sense. We write this model as L ρ (2) . In a similar fashion, we have R(X, ZG + K ) = R n , R(TF ) = R(F), by choosing U suitably.…”

Spectral methods in linear minimax estimation

“…72-74;1990a, pp. 484-485;1990c) is a linearly sufficient statistic (Baksalary andKala, 1981, Drygas, 1983). According to the linear Rao-Blackwell theorem, for any linear statistic Ly there exists a linear statistic Sz such that STy has the same expectation as Ly and possibly a smaller covariance matrix (in the sense of Loewner ordering), namely simply the BLUE (Best Linear Unbiased Estimator) of E(Ly) = LX/3 which can be expressed as a linear function of Ty (STy can also be expressed as BLUP -Best Linear Unbiased Predictor (Drygas, 1983)).…”

“…The criterion given in Baksalary and Kala (1981) and Drygas (1983) is that Ty is linearly sufficient iff Im(X) C_ Im(T') or, equivalently, T -I (0) C_ X '-I (0).…”

Different kinds of sufficiency in the general Gauss-Markov model

“…In connection with the general linear Gauss-Markov model GM (y, Xβ, σ 2 V ), there are some modifications of this notion such as linear sufficiency (B a k s a l a r y and K a l a [2], D r y g a s [3]), invariant linearly sufficiency (O k t a b a , K o r n a c k i , W a w r z o s e k [5]) and quadratic sufficiency ( M u e l l e r [4]). All these variants denote such transformations of the model GM that preserve properties essential in statistical inference.…”