On a matrix result in comparison of linear experiments

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1 Linear Algebra and its Applications 32 (2000) On a matrix result in comparison of linear experiments Czesław Stȩpniak, Institute of Mathematics, Pedagogical University of Rzeszów, Rzeszów, Poland Received 7 January 999; accepted 5 January 2000 Submitted by G.P.H. Styan Abstract A useful statistical relation between normal linear experiments N(Aβ, σ V ) and N(Bβ, σw) is reduced in [Stȩpniak, Statistics 30 (998) ] to the matrix conditions A (V + AA ) A B (W + BB ) B 0andr(V + AA ) r(a) r(w + BB ) r(b).theaim of this note is to present a characterization of these conditions in terms of linear transformations. This characterization may be useful for matrix theory and practice Elsevier Science Inc. All rights reserved. AMS classification: Primary 62J06; Secondary 62B5; 62H20 Keywords: Normal linear experiment; Unbiased estimation; Comparison of experiments; Linear transformarion; White noise. Notation and statistical background The classical vector matrix notation is used here. Among others, if M is a matrix, then M, R(M), r(m) and P M denote, respectively, the transposition, the range (column space), the rank of M and the orthogonal projector on R(M). ByM and Present address: Institute of Applied Mathematics, Agricultural University of Lublin, P.O. Box 58, Akademicka 3, PL Lublin, Poland. Fax: address: stepniak@ursus.ar.lublin.pl (C. Stȩpniak). This work was stimulated by a question concerning Proposition 5 in [2]. I am grateful to Professor Herbert Heyer who asked me this question in a private communication [] /00/$ - see front matter 2000 Elsevier Science Inc. All rights reserved. PII:S (00)

2 322 C. Stȩpniak / Linear Algebra and its Applications 32 (2000) M +, respectively, is denoted a generalized inverse and the Moore Penrose generalized inverse of M. The symbol M 0 means that M is nonnegative definite (n.n.d.). Let us consider two random vectors X and Y of dimensions n and m, respectively. Suppose X is subject to a normal linear model N(Aβ, σ V ) and Y is subject to a normal linear model N(Bβ, σ W), wherea, B, V and W are known matrices, while β R p and σ>0are unknown parameters. In this context X is said to be at least as good as Y if for any parametric function ψ = k β + cσ and for any unbiased estimator ˆψ = ˆψ(Y ), whenever such exists, there exists an unbiased estimator ψ = ψ(x) such that var( ψ) var( ˆψ) for all β and σ. Since the minimal sufficient statistics in the models N(Aβ, σ V ) and N(Bβ, σ W) are complete, thus, by [2,3], X is at least as good as Y if and only if A (V + AA ) A B (W + BB ) B 0 (.) and r(v + AA ) r(a) r(w + BB ) r(b). (.2) A characterization of these conditions in terms of linear transformations plays an important role in comparison of linear experiments and may be useful for matrix theory and practice. A corresponding result given in [2, Proposition 5] includes a misprint and its proof is incomplete. The aim of this note is to present this result in a selfcontained algebraic form and to complete its proof. 2. Main result The following theorem is the main result in this paper. Theorem 2.. For arbitrary n p matrix A and m p matrix B and for arbitrary symmetric n.n.d. matrices V and W such that R(A) R(V ) and R(B) R(W) the following are equivalent: (i) A V A B W B is n.n.d. and r(v) r(a) r(w) r(b). (ii) There exists an m n matrix F such that B = FA and a symmetric n.n.d. matrix G of order m such that R(G) R(FV ) and FVF + G = W. (2.) (iii) There exists an m n matrix F such that B = FA and a symmetric n.n.d. matrix H of order n such that R(H) R(V ) and F(V + H)F = W. (2.2) This theorem is based on the following lemma. Lemma 2.2. For arbitrary n p matrix A and m p matrix B the following are equivalent:

3 C. Stȩpniak / Linear Algebra and its Applications 32 (2000) (a) A A B B is n.n.d. and n r(a) m r(b). (b) There exists an m n matrix F such that B = FAand a symmetric n.n.d. matrix G of order m such that R(G) R(F) and FF + G = I m. (2.3) (c) There exists an m n matrix F such that B = FAand a symmetric n.n.d. matrix H of order n such that F(I n + H)F = I m. (2.4) Remark 2.. If F is any matrix satisfying either (b) or (c) then r(f) = m. Lemma 2.2 is a correct version of Proposition 3 in [2], where the condition R(G) R(F) is missing. For the proof of this lemma we refer to [2]. Here is a complete proof of Theorem 2.. Proof. Since R(A) R(V ) and R(B) R(W), the expressions A V A and B W B do not depend on the choice of generalized inverses V and W. Thus, without loss of generality, one can restrict to the Moore Penrose generalized inverses V + and W +. Let V = n i= λ iv i v i and W = m i= ρ iw i w i be spectral decompositions of the matrices V and W. Wedefine λ v ρ w V =, W =, v λn n ρm w m λ v V 2 = λn v n, W 2 = ρ w ρm w m, A = V A and B = W B. Then A V + A = A A, B W + B = B B, r(v) = n, r(w) = m, r(a ) = r(a), r(b ) = r(b), V 2 V 2 = V, W 2 W 2 = W, and, moreover, V VV = V 2V = I n and W WW = W 2W = I m (2.5) while P V = V 2 V and P W = W 2 W are the orthogonal projectors on R(V ) and R(W), respectively.

4 324 C. Stȩpniak / Linear Algebra and its Applications 32 (2000) Now the condition (i) may be presented in the form (a ) A A B B is n.n.d. and n r(a ) m r(b ). On the other hand, by Lemma 2.2, the condition (a ) is equivalent to any of the following: (b ) There exists an m n matrix F such that B = F A and a symmetric n.n.d. matrix G of order m such that R(G) R(F ) and F F + G = I m. (c ) There exists an m n matrix F such that B = F A and a symmetric n.n.d. matrix H of order n such that F (I n + H)F = I m. Thus it remains to prove that (b ) (ii) and (c ) (iii). (ii) (b ): First note that if F and G satisfy (ii) then P W F and P W G also satisfy the condition. Thus, without loss of generality, one can assume that P W F = F and P W G = G. Then, by the identities (2.5), the condition (2.) may be presented in the form W 2 W FV 2 V VV V 2F W W 2 + G = W. Now, by setting W FV 2 = F,sinceV VV = I n,wegetw 2 F F W 2 + G = W. The last, through the left-hand side multiplying by W and the right-hand side multiplying by W, leads to the desired identity F F + G = I m with G = W GW. It remains to verify that R(G ) R(F ) and F A = B. Really, R(G ) = R(W G) R(W FV) = R(F V 2 ) R(F ) and F A = W FV 2 V A = W FA = W B = B. (iii) (c ): Similarly as before, one can assume, without loss of generality, that P W F = F and P V HP V = H.Then(2.2) may be presented as F ( V 2 V VV V 2 + V 2 V HV V 2) F = W, or, as F (I n + H )F = I m,wheref = W FV 2 and H = V HV. Since the identity F A = B was already shown, the desired implication is proved. (b ) (ii): LetF and G satisfy (b ). Then, by the identities (2.5) and by a respective multiplication, we get W 2 F V V 2 V 2V F W 2 + W 2 G W 2 = W. Therefore, FVF + G = W, wheref = W 2 F V and G = W 2 G W 2. Moreover, FA = W 2 F V A = W 2 F A = W 2 B = W 2 W B = B and R(G) = R(W 2 G ) R(W 2 F ) = R(W 2 F V V 2 ) = R(FV 2 ) = R(FV 2 V 2F ) = R(FV F ) = R(FV ). (c ) (iii): LetF and G satisfy (c ).Then W 2 F V V 2 (I n + H )V 2 V F W 2 = W. Therefore, F(V + H)F = W, wheref = W 2 F V and H = V 2 H V 2.Moreover FA = B and R(H) = R(V 2 H ) R(V 2 ) = R(V 2 V 2) = R(V ). This completes the proof of Theorem 2..

5 C. Stȩpniak / Linear Algebra and its Applications 32 (2000) References [] H. Heyer, (998), Private communication. [2] C. Stȩpniak, Comparing normal linear experiments and transformation of observations, Statistics 30 (998) [3] C. Stȩpniak, S.G. Wang, C.F.J. Wu, Comparison of linear experiments with known covariances, Ann. Statist. 2 (984)