Estimation under a general partitioned linear model

Transcription

1 Linear Algebra and its Applications 321 (2000) Estimation under a general partitioned linear model Jürgen Groß a,, Simo Puntanen b a Department of Statistics, University of Dortmund, Vogelpothsweg 87, D Dortmund, Germany b Department of Mathematics, Statistics and Philosophy, University of Tampere, P.O. Box 607, FIN Tampere, Finland Received 7 November 1998; accepted 5 January 2000 Submitted by G.P.H. Styan Abstract In this paper, we consider a general partitioned linear model and a corresponding reduced model. We derive a necessary and sufficient condition for the BLUE for the expectation of the observable random vector under the reduced model to remain BLUE in the partitioned model. The former is shown to be always an admissible estimator under a mild condition. We also regard alternative linear estimators and their coincidence with the BLUE under the partitioned model Elsevier Science Inc. All rights reserved. AMS classification: 62J05; 62H12 Keywords: Partitioned linear model; Best linear unbiased estimation; Admissible estimation; Orthogonal projector 1. Introduction Let R m,n denote the set of m n real matrices. The symbols A, A +, A, C(A), N(A),andrk(A) will stand for the transpose, the Moore Penrose inverse, any generalized inverse, the column space, the null space, and the rank, respectively, of A R m,n.bya we denote any matrix satisfying C(A ) = N(A ).Further,P A = AA + Corresponding author. Fax: addresses: gross@amadeus.statistik.uni-dortmund.de (J. Groß), sjp@uta.fi (S. Puntanen) /00/$ - see front matter 2000 Elsevier Science Inc. All rights reserved. PII:S (00)

2 132 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) denotes the orthogonal projector (with respect to the standard inner product) onto C(A),andM A = I P A. In particular, we denote P i = P Xi, M i = I P i,i= 1, 2. Consider a general Gauss Markov model denoted by M ={y, Xβ,σ 2 V}, E(y) = Xβ, D(y) = σ 2 V, (1.1) where X is a known n p matrix, β a p 1 vector of unknown parameters, V a known n n nonnegative definite matrix, and σ 2 > 0 is an unknown scalar. E( ) and D( ) denote expectation and dispersion of a random vector argument. It is assumed that the model is consistent,thatis y C(X : V), (1.2) see [6,14,15]. A vector of parametric functions Kβ, wherek R k,p, is estimable under the model M if and only if K = CX for some C R k,n. It is well known, see e.g. [16, p. 282], that under the model M the best linear unbiased estimator (BLUE) for an estimable vector of parametric functions CXβ is given by Fy,whereF R k,n is any solution to F(X : VX ) = C(X : 0). (1.3) We may also conclude that if Gy is the BLUE for Xβ, i.e., the matrix G R n,n is a solution to G(X : VX ) = (X : 0), (1.4) then CG is a solution to (1.3), and therefore CGy is BLUE for CXβ. By partitioning X = (X 1 : X 2 ) so that X 1 has p 1 columns and X 2 has p 2 columns with p = p 1 + p 2, and by accordingly writing β = (β 1, β 2 ), we can express M in its partitioned form M ={y, X 1 β 1 + X 2 β 2,σ 2 V}. (1.5) Regarding β 1 as a nuisance parameter, our interest focuses on estimation of a vector of estimable parametric functions K 2 β 2. Lemma 1. Under the model M ={y, X 1 β 1 + X 2 β 2,σ 2 V}, the vector of parametric functions K 2 β 2 is estimable if and only if K 2 = C 2 M 1 X 2 (1.6) for some matrix C 2, where M 1 = I P 1 is the orthogonal projector onto N(X 1 ). Proof. The vector K 2 β 2 is estimable under the model M if and only if (0 : K 2 ) = C(X 1 : X 2 ) (1.7) for some C. IfK 2 = C 2 M 1 X 2,thenC = C 2 M 1 satisfies (1.7). Conversely, if (1.7) holds for some C, thencx 1 = 0, implying C = C 2 M 1 for some C 2. Hence K 2 = C 2 M 1 X 2.

3 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) Reduced models In view of Lemma 1, our interest is lead to estimation of M 1 X 2 β 2. Under model M, the linear transform M 1 y of the observable random vector y has expectation E(M 1 y) = M 1 X 2 β 2 and dispersion D(M 1 y) = σ 2 M 1 VM 1. Hence, we obtain a reduced linear model M cr ={M 1 y, M 1 X 2 β 2,σ 2 M 1 VM 1 }, (2.1) which is in accordance with model M and is appropriate for inferences about M 1 X 2 β 2.Suchacorrectly reduced model has been considered for example in [3 5,10 12]. On the other hand, the triplet {y, M 1 X 2 β 2,σ 2 V} contains all information, which we need for estimating M 1 X 2 β 2. Hence, we can raise the question whether it is possible to obtain estimators for M 1 X 2 β 2 by regarding the triplet {y, M 1 X 2 β 2,σ 2 V} as a reduced model M r ={y, M 1 X 2 β 2,σ 2 V}, E(y) = M 1 X 2 β 2, D(y) = σ 2 V. (2.2) Such a model has been considered by Bhimasankaram and Saha Ray [3] and Bhimasankaram et al. [5]. It should be emphasized that we do not propose to consider model M r for practical purposes. For this, model M cr would probably be the better choice in most cases. As a matter of fact, one of the referee s pointed out that using model M r in practice could be quite obscure. Therefore we rather like to think of model M r as a source of estimators whose properties under the true model M are investigated in Sections 3 5. Eventually in Section 6 we reconsider the correctly reduced model M cr. Let us now turn our attention to model M r. Clearly, if we consider M r as a linear model, then we have to assume that y C(M 1 X 2 : V) with probability one. (2.3) On the other hand, it would not contradict our original inference base under the model M if y realizes in C(X 1 : X 2 : V) but not in C(M 1 X 2 : V). Inotherwords, inconsistency of model M r, i.e., y / C(M 1 X 2 : V), does not automatically imply inconsistency of model M. To overcome any logical difficulties which arise from regarding a reduced model M r, we assume that the subspace in which y realizes almost surely is the same under both models, i.e., C(X 1 : X 2 : V) = C(M 1 X 2 : V). (2.4) In that case we will call M not contradictory to M r. Obviously, if M is not contradictory to M r, then consistency of M implies consistency of M r in view of C(M 1 X 2 : V) C(X 1 : X 2 : V). If model M is only weakly singular,thatis C(X 1 : X 2 ) C(V), (2.5) then M is never contradictory to M r. In the following lemma we collect together some properties related to condition (2.4).

4 134 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) Lemma 2. Let X 1 R n,p1, X 2 R n,p2, and let V R n,n be nonnegative definite. Then: (i) The following three conditions are equivalent: (a1) C(X 1 : X 2 : V) = C(M 1 X 2 : V), (a2) C(X 1 ) C(M 1 X 2 : V), (a3) rk(x 1 ) + dim[c(m 1 X 2 ) C(V)] =dim[c(x 1 : X 2 ) C(V)]. (ii) Condition (a1) implies the following four conditions (three first being equivalent): (b1) C(X 1 ) = C(P 1 V), (b2) rk(x 1 ) = rk(vx 1 ), (b3) C(X 1 ) C(V ) ={0}, (b4) C(X 1 ) C[(I P M1 X 2 )V]. (iii) The following two conditions are equivalent: (c1) C(X 1 ) C(V), (c2) C(X 1 ) [C(M 1 X 2 ) C(V)] =C(X 1 : X 2 ) C(V). (iv) Furthermore, (d1) condition (c1) implies (a1), (d2) condition (a1) does not imply (c1). Proof. In view of C(X 1 : X 2 : V) = C(X 1 : X 2 ) + C(V) = C(X 1 ) + C(M 1 X 2 ) + C(V), (cf. e.g. [9]), condition (a1) is equivalent to (a2). Since always C(M 1 X 2 : V) C(X 1 : X 2 : V), (a1) is equivalent to rk(x 1 : X 2 : V) = rk(m 1 X 2 : V). (2.6) From the identities rk(m 1 X 2 : V)=rk(M 1 X 2 ) + rk(v) dim[c(m 1 X 2 ) C(V)], rk(x 1 : X 2 : V)=rk(X 1 : X 2 ) + rk(v) dim[c(x 1 : X 2 ) C(V)], (and from rk(x 1 : X 2 ) = rk(x 1 ) + rk(m 1 X 2 )) we see that (2.6) is equivalent to (a3). Thus part (i) of the lemma is proved. Assume for part (ii) that (a2) holds. Then there exist matrices A and B such that X 1 = M 1 X 2 A + VB. (2.7) Premultiplying (2.7) by P 1 yields X 1 = P 1 VB, i.e., C(X 1 ) C(P 1 V). (2.8)

5 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) Since C(P 1 V) C(X 1 ), (2.8) is equivalent to rk(x 1 ) = rk(p 1 V) = rk(x 1 V) = rk(vx 1). (2.9) In light of [9], (2.9) holds if and only if rk(x 1 ) = rk(x 1 V) = rk(x 1) dim[c(x 1 ) C(V )], i.e., if and only if (b3) holds. If (2.7) is premultiplied by I P M1 X 2, then we obtain X 1 = (I P M1 X 2 )VB, thus showing that (a1) indeed implies (b4). This shows part (ii) of the lemma. For part (iii) let (c1) be satisfied. It is obvious that (c1) implies (a1) and hence (a3). But since C(X 1 ) C(V) is equivalent to C(X 1 ) = C(X 1 ) C(V), and since always [C(X 1 ) C(V)] [C(M 1 X 2 ) C(V)] C(X 1 : X 2 ) C(V), (2.10) where C(X 1 : X 2 ) = C(X 1 ) C(M 1 X 2 ), (a3) is equivalent to (c2), showing that (c1) implies (c2). Conversely, if (c2) holds, then C(X 1 ) C(X 1 : X 2 ) C(V) C(V), i.e., (c1). Hence part (iii) is shown. For part (iv) we note that (d1) has already been mentioned above. To prove (d2), take (X 1 : X 2 ) = 0 1, V = (2.11) Then (a1) holds but C(X 1 ) is not contained in C(V). We note that statement (d2) is in contradiction with a statement by Bhimasankaram et al. [5, Section 1]. Unfortunately, the aforementioned authors seem to believe that condition (a1) is equivalent to condition (c1), which is easily disproved by (2.11). In Section 3, we investigate conditions under which the BLUE for M 1 X 2 β 2 under model M r remains BLUE under the partitioned model M, where it is assumed that M is not contradictory to M r. Note that Bhimasankaram and Saha Ray [3, Theorem 2.4] investigate a similar problem when V is positive definite by supplying a sufficient condition for coincidence of both BLUEs. Bhimasankaram et al. [9, Theorem 3.1] study a generalization to the case of a singular matrix V such that C(X 1 ) C(V). 3. Best linear unbiased estimation The following two lemmas give characterizations of the BLUEs of M 1 X 2 β 2 under models M and M r, respectively. Since model M is assumed to be not contradictory to model M r, every two representations F 1 y and F 2 y of the BLUE of M 1 X 2 β 2 under model M r, where possibly F 1 = F 2, satisfy F 1 y = F 2 y for all y C(M 1 X 2 : V) = C(X 1 : X 2 : V). (3.1)

6 136 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) This means, if we consider the set of different representations of the BLUE for M 1 X 2 β 2 under M r as a set of linear estimators for M 1 X 2 β 2 under M, thenall these estimators coincide almost surely under M, provided the model M is not contradictory to the model M r. This is true even if no BLUE for M 1 X 2 β 2 under M r remains BLUE under M. Lemma 3. Let Z = I P M1 X 2, and let V = M 1 VM 1. The following four statements are equivalent: (i) Fy is BLUE for M 1 X 2 β 2 under the model M ={y, X 1 β 1 + X 2 β 2,σ 2 V}. (ii) F satisfies F(X 1 : X 2 ) = (0 : M 1 X 2 ) and FVM 1 Z = 0. (iii) F = NM 1,whereN satisfies NM 1 X 2 = M 1 X 2 and NV Z = 0. (iv) F =[I V Z(ZV Z) + Z]M 1 + P[I ZV Z(ZV Z) + ]ZM 1 for some P. Proof. An estimator Fy is BLUE for M 1 X 2 β 2 under the model M if and only if F(X 1 : X 2 ) = (0 : MX 2 ) and FV(X 1 : X 2 ) = 0, where (X 1 :X 2 ) is any matrix satisfying C[(X 1 :X 2 ) ]=N[(X 1 : X 2 ) ]. But since M 1 Z = M 1 (I P M1 X 2 ) = M 1 P M1 X 2 = I P (X1 :X 2 ), equivalence between (i) and (ii) is shown. It is clear that (iii) implies (ii). Conversely, if (ii) is satisfied, then FX 1 = 0 implies F = NM 1 for some N and (iii) follows. From [17, Theorem 1] we know that the general solution to the equations NM 1 X 2 = M 1 X 2 and NV Z = 0 with respect to N is N =[I V Z(ZV Z) + Z]+A[I ZV Z(ZV Z) + ]Z for arbitrary A. Hence equivalence between (iii) and (iv) holds. Note that we will not need condition (iv) of Lemma 3 in this paper. The BLUE under the reduced model can be characterized as follows. Lemma 4. Let Z = I P M1 X 2. The following three statements are equivalent: (i) Fy is BLUE for M 1 X 2 β 2 under the model M r ={y, M 1 X 2 β 2,σ 2 V}. (ii) F satisfies FM 1 X 2 = M 1 X 2 and FVZ = 0. (iii) F =[I VZ(ZVZ) + Z]+B[I ZVZ(ZVZ) + ]Z for some B. Proof. Equivalence between (i) and (ii) follows immediately by noting that Z is the orthogonal projector onto N(X 2 M 1), and therefore is a special choice for (M 1 X 2 ). Equivalence between (ii) and (iii) follows from [17, Theorem 1]. We may now state the main result of this section. Theorem 1. Let the partitioned model M ={y, X 1 β 1 + X 2 β 2,σ 2 V} be not contradictory to the reduced model M r. Then every BLUE for M 1 X 2 β 2 under M r = {y, M 1 X 2 β 2,σ 2 V} remains BLUE for M 1 X 2 β 2 under M if and only if

7 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) C(X 1 ) C[V(I P M1 X 2 )]. (3.2) Proof. Suppose that Fy is BLUE for M 1 X 2 β 2 under M r, meaning that F is any matrix from Lemma 4. By Lemma 4 (ii) it follows FV = FVP M1 X 2.SinceP M1 X 2 = P M1 X 2 M 1,thisgives FV = FVM 1. (3.3) Hence from condition (ii) of Lemma 3, Fy is BLUE under M if and only if FX 1 = 0 and FX 2 = M 1 X 2. (3.4) But FX 1 = 0 if and only if FM 1 = F, sothatfx 1 = 0 implies FX 2 = FM 1 X 2 = M 1 X 2, where the latter equality comes from Lemma 4 (ii). This shows that Fy,being the BLUE under M r, is BLUE under M if and only if FX 1 = 0. (3.5) It remains to show that (3.2) is equivalent to (3.5) for every BLUE Fy under M r. Suppose that the latter is satisfied, i.e., (3.5) holds for every F from condition (iii) in Lemma 4. Then [I VZ(ZVZ) + Z]X 1 = 0, showing (3.2). Conversely if (3.2) holds, then C(ZX 1 ) C(ZVZ), i.e., ZVZ(ZVZ) + ZX 1 = ZX 1. Hence from Lemma 4 (iii), FX 1 =[I VZ(ZVZ) + Z]X 1 (3.6) for every BLUE Fy under M r. But in view of (3.2), X 1 = VZA for some A. Since clearly VZ(ZVZ) + ZVZ = VZ, we arrive at FX 1 = 0, thus concluding the proof. Remark 1. From the proof of Theorem 1 it becomes evident that the assertion remains true when the phrase remains BLUE for M 1 X 2 β 2 under M is replaced by is unbiased for M 1 X 2 β 2 under M. In connection with a weakly singular model M, we may state a result which is in accordance with the equivalence of (a) and (c) in [12, Theorem 1]. Corollary 1. Let the partitioned model M be weakly singular. Then every BLUE for M 1 X 2 β 2 under the reduced model M r remains BLUE for M 1 X 2 β 2 under M if and only if X 2 M 1V X 1 = 0 (3.7) for any generalized inverse V of V. Proof. In advance we note that the invariance of X 2 M 1V X 1 with respect to the choice of generalized inverse V is equivalent to C(X 1 ) C(V) and C(M 1 X 2 ) C(V), (3.8) cf. [18, p. 43]. Of course it is assumed that X 1 = 0 and X 2 M 1 = 0. Conditions (3.8) are equivalent to C(X 1 : X 2 ) C(V), which means that M is weakly singular.

8 138 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) If (3.7) is satisfied, then obviously V X 1 = ZA for some matrix A, wherez = I P M1 X 2. In addition, the weak singularity implies VV X 1 = X 1 and hence X 1 = VZG. Thus from Theorem 1 every BLUE for M 1 X 2 β 2 under M r remains BLUE for M 1 X 2 β 2 under M. Conversely, if condition (3.2) from Theorem 1 is satisfied, i.e., C(X 1 ) C(VZ), andm is weakly singular, i.e., C(X 1 : X 2 ) C(V), then X 2 M 1V X 1 = X 2 M 1V VZA = X 2 M 1ZA = 0 for some matrix A and every generalized inverse V of V. Obviously, when C(X 1 ) C(V), then we can never expect that the BLUE for M 1 X 2 β 2 under M r remains BLUE under M. On the other hand, when C(X 1 ) = C(VX 1 ), then condition (3.2) is always satisfied in view of X 1 = ZX 1. When the partitioned model M is not contradictory to M r, then the condition C(X 1 ) = C(VX 1 ) is equivalent to C(VX 1 ) C(X 1 ), as shown in the proof of the following corollary. Corollary 2. Let the partitioned model M be not contradictory to the reduced model M r. Then every BLUE for M 1 X 2 β 2 under M r remains BLUE for M 1 X 2 β 2 under M if C(VX 1 ) C(X 1 ). (3.9) Proof. If model M is not contradictory to M r, then condition (b2) of Lemma 2 implies rk(x 1 ) = rk(vx 1 ). Hence, (3.9) is equivalent to C(X 1 ) = C(VX 1 ), and therefore C(X 1 ) = C(VZX 1 ) C(VZ). This shows that condition (3.2) from Theorem 1 is satisfied under (3.9). It follows easily from Lemmas 3 (ii) and 4 (ii) that under C(VX 1 ) C(X 1 ) every BLUE for M 1 X 2 β 2 under M remains BLUE for M 1 X 2 β 2 under M r.inother words, if C(VX 1 ) C(X 1 ), then the sets of BLUEs under M and M r, respectively, coincide. We note that the condition C(VX 1 ) C(X 1 ) is necessary and sufficient for equality of ordinary least squares estimator (OLSE) and BLUE for X 1 β 1 under a linear model {y,e(y) = X 1 β 1,D(y) = σ 2 V}, cf. [13]. Corollary 2 has also been established by Bhimasankaram and Saha Ray [3, Theorem 2.4] and Bhimasankaram et al. [5, Theorem 3.1] under more restrictive assumptions. We will now consider the estimator P M1 X 2 y as the OLSE for M 1 X 2 β 2.Ifthis estimator is BLUE under M r, then it remains BLUE under M, as demonstrated by the following corollary. Corollary 3. Let the partitioned model M be not contradictory to the reduced model M r.ifp M1 X 2 y is the BLUE for M 1 X 2 β 2 under the reduced model M r, then it remains BLUE under M.

9 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) Proof. Condition (b4) of Lemma 2 implies that C(X 1 ) C(ZV), wherez = I P M1 X 2.ButifP M1 X 2 y is BLUE for M 1 X 2 β 2 under the reduced model M r,then from [13], VZ = ZV. Hence, C(X 1 ) C(ZV), showing that condition (3.2) from Theorem 1 is satisfied. 4. Admissible estimation When the BLUE for M 1 X 2 β 2 under model M r does not remain BLUE under model M, it is natural to ask whether this estimator would make any sense under the partitioned model M. In this section, we give a partial answer to this question by demonstrating that there cannot exist a linear estimator which is uniformly better, provided C(X 1 ) C(V). More precisely, when C(X 1 ) C(V), then the BLUE for M 1 X 2 β 2 under model M r is an admissible estimator for M 1 X 2 β 2 among the set of linear estimators L n (y) ={Ly + λ : L R n,n, λ R n,1 } (4.1) for M 1 X 2 β 2 under the partitioned model M. According to Baksalary and Markiewicz [2] a linear estimator Ay + a is called admissible among L n (y) under M if there does not exist Ly + λ L n (y) such that the inequality ϱ(ly + λ; M 1 X 2 β 2 ) ϱ(ay + a; M 1 X 2 β 2 ) holds for every pair (β,σ 2 ) Θ and is strict for at least one pair (β,σ 2 ) Θ,whereΘ is the parameter space corresponding to model M,and ϱ(ly + λ; M 1 X 2 β 2 ) = E[(Ly + λ M 1 X 2 β 2 ) (Ly + λ M 1 X 2 β 2 )] (4.2) is the quadratic risk of Ly + λ L n (y) under M. The following lemma, which follows easily from the main result in [2, Theorem], characterizes homogenous linear admissible estimators for M 1 X 2 β 2 under model M. Lemma 5. An estimator Ay is admissible for M 1 X 2 β 2 among L n (y) under the partitioned model M if and only if A R n,n satisfies the following four conditions: C(VA ) C(X 1 : X 2 ), (4.3) AVM 1 is symmetric, (4.4) AV(M 1 A ) is nonnegative definite, (4.5) C[(A M 1 )(X 1 : X 2 )]=C[(A M 1 )W], (4.6) where W is any matrix such that C(W) = C(X 1 : X 2 ) C(V). The following result shows that the BLUE for M 1 X 2 β 2 under the reduced model M r can be regarded as a reasonable choice of estimator under the partitioned

10 140 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) model M, provided C(X 1 ) C(V), since it cannot be uniformly outperformed by a different linear estimator. Theorem 2. Let the partitioned model M ={y, X 1 β 1 + X 2 β 2,σ 2 V} be not contradictory to the reduced model M r ={y, M 1 X 2 β 2,σ 2 V}. Then every BLUE for M 1 X 2 β 2 under M r is admissible for M 1 X 2 β 2 among L n (y) under M if C(X 1 ) C(V). (4.7) Proof. The assertion is proved when the four conditions (4.3) (4.6) are true for F replacing A,whereFis any matrix satisfying Lemma 4. From Lemma 4(ii) we have FVZ = 0, wherez = I P M1 X 2. This may equivalently be expressed as C(VF ) N(Z),whereN(Z) = C(M 1 X 2 ) C(X 1 : X 2 ). Thus, C(VF ) C(X 1 : X 2 ). (4.8) Moreover, C(VF ) C(M 1 X 2 ) implies FV = FVM 1. In view of the identity ZVZ(ZVZ) + ZV = ZV it follows from Lemma 4(iii) that FV = V VZ(ZVZ) + ZV. Therefore, FVM 1 = V VZ(ZVZ) + ZV is symmetric. (4.9) Since we easily compute FVF = FV, it is seen that FVM 1 FVF = FV FV = 0 is nonnegative definite. (4.10) From Lemma 4(ii) we have (F M 1 )M 1 X 2 = 0. Hence, in view of C(X 1 : X 2 ) = C(X 1 ) C(M 1 X 2 ), and in view of C(W) = C(X 1 ) [C(M 1 X 2 ) C(V)] by Lemma 2 (iii), we obtain C[(F M 1 )(X 1 : X 2 )]=C[(F M 1 )W] =C(FX 1 ). (4.11) The assertion now follows from (4.8) to (4.11) by Lemma Alternative estimation We will now consider estimators for M 1 X 2 β 2 of the form FM 1 y,wheref is any matrix such that Fy is the BLUE for M 1 X 2 β 2 under the reduced model M r.an estimator FM 1 y can be seen as the generalized version of an estimator which has been considered by Aigner and Balestra [1], see also [19] for related results. We pose the question whether an estimator of the form FM 1 y can be reasonably used under the partitioned model M. It is clear that FM 1 y is unbiased for M 1 X 2 β 2 under model M in view of FM 1 (X 1 : X 2 ) = (0 : M 1 X 2 ). (5.1) Obviously, since FM 1 y is unbiased for M 1 X 2 β 2,theBLUEforM 1 X 2 β 2 is uniformly not worse than FM 1 y with respect to the quadratic risk of estimators. Therefore,

11 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) FM 1 y can be admissible for M 1 X 2 β 2 among L n (y) under M only if it coincides with the BLUE for M 1 X 2 β 2 under M. The following theorem gives a necessary and sufficient condition for the latter. Theorem 3. Let the partitioned model M ={y, X 1 β 1 + X 2 β 2,σ 2 V} be not contradictory to the reduced model M r, and let Fy be BLUE for M 1 X 2 β 2 under M r = {y, M 1 X 2 β 2,σ 2 V}. Then every estimator FM 1 y is BLUE for M 1 X 2 β 2 under M if and only if C(P 1 VM 1 Z) C(VZ), (5.2) where Z = I P M1 X 2. Proof. Since for any matrix F from Lemma 4 identity (5.1) holds, it follows by Lemma 3(ii) that FM 1 y is BLUE for M 1 X 2 β 2 under M if and only if FM 1 VM 1 Z = 0. (5.3) Now, let (5.3) be satisfied for every matrix F from Lemma 4(iii). Then (choosing B = 0), [I VZ(ZVZ) + Z]M 1 VM 1 Z = 0. (5.4) In view of M 1 = I P 1 and M 1 Z = ZM 1, (5.4) can be written as [I VZ(ZVZ) + Z](VZM 1 P 1 VM 1 Z) = 0, (5.5) which in view of VZ(ZVZ) + ZVZ = VZ is equivalent to [I VZ(ZVZ) + Z]P 1 VM 1 Z = 0. (5.6) Since it is easily seen that N[I VZ(ZVZ) + Z]=C(VZ), identity (5.6) is equivalent to (5.2). Conversely, let (5.2) be satisfied. As just shown above, (5.2) is equivalent to (5.6), which in turn is equivalent to (5.4). Condition (5.4) also entails [Z ZVZ(ZVZ) + Z]M 1 VM 1 Z = 0. (5.7) Now, (5.4) and (5.7) show that for every matrix F from Lemma 4(iii) condition (5.3) is satisfied and hence FM 1 y is BLUE for M 1 X 2 β 2 under M. Remark 2. Condition (5.2) from Theorem 3 may alternatively be expressed as C(P 1 VM) C(VZ), (5.8) where P 1 = X 1 X + 1, M = I P (X 1 :X 2 ) and Z = I P M1 X 2. The condition C(P 1 VM 1 Z) C(VZ) from Theorem 3 is obviously weaker than the condition C(X 1 ) C(VZ) from Theorem 1, since the latter implies the former. That the converse is not true can be seen from the matrices X 1 = 1 1, X 2 = , V = (5.9)

12 142 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) Then C(P 1 VM 1 Z) C(VZ) but C(X 1 ) C(VZ). The same matrices can be used to demonstrate that [7, Theorem 1] is false. Under a partitioned model M with positive-definite V and (X 1 : X 2 ) of full column rank, the authors claim that a necessary and sufficient condition for equality of the generalized least squares (GLS) estimator and the so-called pseudo-gls estimator for (the unbiasedly estimable vector) β 2 is X 2 M 1V 1 X 1 = 0. However, under (5.9) it is easily computed that the estimators in question coincide without satisfying this condition. The correct condition, being X 2 M 1V + P 1 VM 1 Z = 0, appears in [12, Theorem 1]. It is related to our Theorem 3 in the following way. Corollary 4. Let the partitioned model M be weakly singular, and let Fy be BLUE for M 1 X 2 β 2 under the reduced model M r. Then every estimator FM 1 y is BLUE for M 1 X 2 β 2 under M if and only if X 2 M 1V P 1 VM 1 Z = 0 (5.10) for any generalized inverse V of V, and Z = I P M1 X 2. Proof. From the proof of Corollary 1 we know that X 2 M 1V X 1 is invariant with respect to the choice of generalized inverse V in view of C(X 1 ) C(V) and C(M 1 X 2 ) C(V). (5.11) Hence the left-hand side of (5.10) does not depend on the choice V. From Theorem 3, the assertion is true if (5.10) is equivalent to (5.2). Clearly (5.2) implies (5.10). To go the other way, assume that (5.10) holds. Then V P 1 VM 1 Z = ZA (5.12) for some matrix A. Premultiplying (5.12) by V yields (5.2) since VV P 1 = P Frisch Waugh estimation In Section 2, we introduced the correctly reduced model M cr ={M 1 y, M 1 X 2 β 2,σ 2 M 1 VM 1 }, E(M 1 y) = M 1 X 2 β 2, D(M 1 y) = σ 2 (6.1) M 1 VM 1, which is in accordancewith model M. We state the following result as an immediate consequence of Lemma 3. Theorem 4. Every BLUE for M 1 X 2 β 2 under the correctly reduced model M cr remains BLUE for M 1 X 2 β 2 under the partitioned model M. Proof. An estimator is BLUE for M 1 X 2 β 2 under M cr if and only if it is of the form NM 1 y,wherensatisfies NM 1 X 2 = M 1 X 2 and NM 1 VM 1 Z = 0 with Z = I P M1 X 2. But then from Lemma 3 (iii), NM 1 y is BLUE for M 1 X 2 β 2 under M.

13 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) It is obvious that also the reverse relation holds in Theorem 4, i.e., every BLUE for M 1 X 2 β 2 under M remains BLUE for M 1 X 2 β 2 under M cr. In other words, the sets of BLUEs under M and M cr, respectively, coincide. Estimation under the correctly reduced model M cr as carried out in Theorem 4 can be seen as a generalization of a well-known procedure due to [8] to the case of possibly nonsingular V and possibly nonestimable β 2. Acknowledgement The first author was supported by the Deutsche Forschungsgemeinschaft under grant Tr 253/2-3. This research was done when the second author was a Senior Scientist of the Academy of Finland. References [1] D.J. Aigner, P. Balestra, Optimal experimental design for error components models, Econometrica 56 (1988) [2] J.K. Baksalary, A. Markiewicz, Admissible linear estimators in the general Gauss Markov model, J. Statist. Plan. Inf. 19 (1988) [3] P. Bhimasankaram, R.S. Ray, On a partitioned linear model and some associated reduced models, Linear Algebra Appl. 264 (1997) [4] P. Bhimasankaram, D. Sengupta, The linear zero functions approach to linear models, Sankhya Ser. B 58 (1996) [5] P. Bhimasankaram, K.R. Shah, R.S. Ray, On a singular partitioned linear model and some associated reduced models, J. Combin. Inform. Systems Sci. (to appear). [6] A. Feuerverger, D.A.S. Fraser, Categorial information and the singular linear model, Canadian J. Statist. 8 (1980) [7] D.G. Fiebig, R. Bartels, W. Krämer, The Frisch Waugh theorem and generalized least squares, Econometric Rev. 15 (1996) [8] R. Frisch, F.V. Waugh, Partial time regressions as compared with individual trends, Econometrica 1 (1933) [9] G. Marsaglia, G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra 2 (1974) [10] M. Nurhonen, S. Puntanen, A property of partitioned generalized regression, Commun. Statist. Theory Meth. 21 (1992) [11] S. Puntanen, Some matrix results related to a partitioned singular linear model, Commun. Statist. Theory Meth. 25 (1996) [12] S. Puntanen, Some further results related to reduced singular linear models, Commun. Statist. Theory Meth. 26 (1997) [13] S. Puntanen, G.P.H. Styan, The equality of the ordinary least squares estimator and the best linear unbiased estimator with discussion, Amer. Statist. 43 (1989) [14] C.R. Rao, Unified theory of linear estimation, Sankhya, Ser. A 33 (1971) [15] C.R. Rao, Corrigenda, Sankhya, Ser. A 34 (1972) 194, 477. [16] C.R. Rao, Representations of best linear unbiased estimators in the Gauss Markoff model with a singular dispersion matrix, J. Multivariate Anal. 3 (1973)

14 144 J. Groß, S. Puntanen / Linear Algebra and its Applications 321 (2000) [17] C.R. Rao, Choice of best linear estimators in the Gauss Markoff model with a singular dispersion matrix, Commun. Statist. Theory Meth. 7 (1978) [18] C.R. Rao, S.K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley, New York, [19] H.J. Werner, C. Yapar, More on partitioned possibly restricted linear regression, in: E.-M. Tiit, T. Kollo, H. Niemi (Eds.), Multivariate Statistics and Matrices in Statistics, VSP, Utrecht, Netherlands, TEV, Vilnius, Lithuania, 1995, pp