A mixed model Yo = ∑m i=1 Xiβi + ∑i=m+1 w Xiβ̃ + e is orthogonal when the matrices Mi = XiXi T, i = 1,...,w, commute. The vectors βc1 1,...,βcm m are fixed vectors and the βcm+1 m+1,...,βcw w and en are random. For these models we have very interesting results namely we have UMVUE for the relevant parameters when normality is assumed. We now intend to generalize that class of models taking Y = L(∑i=1 mXiβi + ∑i=m+1 w Xiβ̃ + e with L a matrix whose column vectors are linearly independent.authorsversionpublishe

Ferreira, Dário

Ferreira, Sandra Saraiva

Mexia, João Tiago

Moreira, Elsa

Journal of Interdisciplinary Mathematics

New University of Lisbon's Repository

Inference for L orthogonal models

Repositório da Universidade Nova de Lisboa

Full Terms & Conditions of access and use can be found athttps://www.tandfonline.com/action/journalInformation?journalCode=tjim20Journal of Interdisciplinary MathematicsISSN: 0972-0502 (Print) 2169-012X (Online) Journal homepage: https://www.tandfonline.com/loi/tjim20Inference for L orthogonal modelsSandra Saraiva Ferreira , Dário Ferreira , Elsa Moreira & João Tiago MexiaTo cite this article: Sandra Saraiva Ferreira , Dário Ferreira , Elsa Moreira & João Tiago Mexia(2009) Inference for L orthogonal models, Journal of Interdisciplinary Mathematics, 12:6, 815-824,DOI: 10.1080/09720502.2009.10700666To link to this article:  https://doi.org/10.1080/09720502.2009.10700666Published online: 28 May 2013.Submit your article to this journal Article views: 27Citing articles: 2 View citing articles Inference for L orthogonal modelsSandra Saraiva Ferreira ∗2Da´rio FerreiraDepartment of Mathematics4University of Beira InteriorAv. Marqueˆs d’ A´vila e Bolama66200 Covilha˜Portugal8Elsa MoreiraJoa˜o Tiago Mexia10Department of MathematicsFaculty of Science and Technology12New University of Lisbon2829 Caparica14PortugalAbstract16A mixed model Yo = ∑mi=1 Xiβi + ∑wi=m+1 Xiβ˜i + e is orthogonal when the matricesMi = XiXTi , i = 1, . . . , w, commute. The vectors βc11 , . . . ,βcmm are fixed vectors and the18βcm+1m+1 , . . . ,βcww and en are random. For these models we have very interesting results namelywe have UMVUE for the relevant parameters when normality is assumed. We now intend to20generalize that class of models taking Y = L(∑mi=1 Xiβi +∑wi=m+1 Xiβ˜i)+ e with L a matrixwhose column vectors are linearly independent.22Keywords and phrases : Normal orthogonal models, commutative Jordan algebras, mixed model.1. Introduction24An L model is given byY = LYo + e , (1.1)26∗E-mail: sandra@mat.ubi.pt——————————–Journal of Interdisciplinary MathematicsVol. 12 (2009), No. 6, pp. 815–824c© Taru Publications816 S. S. FERREIRA ET AL.whereYo =m∑i=1Xiβi +w∑i=m+1Xiβ˜i (1.2)2in the core model and e is the error vector. The vectors βc11 , . . . ,βcmm arefixed vectors and the βcm+1m+1 , . . . ,βcww and en are independent, with null4mean vectors and covariance matrices σ2i Ici , i = m + 1, . . . , w and σ2 In .Then Yo has mean vector and covariance matrix6uo =m∑i=1Xiβi (1.3)and8Vo =w∑i=m+1σ2i Mi +σ2L+L+>. (1.4)We will assume that the column vectors of matrix L are independent and10that the n˙ components of core model correspond to the treatments. Forinstance12L = D(1r1 , . . . , 1rn˙) (1.5)is the blockwise diagonal matrix with as principal blocks the vectors14whose r1, . . . , rn˙ components are equal to one. We will use a mixed modelwith those number of observations for the different treatments.16We will be interested in the case in which the core model is orthogonalassociated to a commutative Jordan algebra, CJA, A . Then, see [1], the18matrices Mi = XiXTi , i = 1, . . . , w, commute and constitute a basis for A .In the next section we will consider CJA presenting certain results20that will be useful later on. Then we study the structure of the core modelbefore presenting inference.222. Commutative Jordan algebras and core modelsCommutative Jordan algebra, CJA, are linear spaces constituted by24symmetric matrices that commute and containing the squares of its matri-ces. These structures were introduced by [3], in a generalized formulation26for quantum mechanics. Later on, CJA were undiscovered and used inlinear statistical inference by [10], [11], [12] and [14], among others like [13]28and by [15] and [16]. Namely, [12] showed that for each CJA, A , there isone and only one basis, the principal basis, pb(A) , constituted by pairwise30orthogonal projection matrices which are orthogonal. An example of CJAis the family V(P) of symmetric matrices diagonalized by P .32L ORTHOGONAL MODELS 817The symmetric matrices M1, . . . , Mu commute, see [9], if and only ifthey are diagonalized by the same orthogonal matrix P thus belonging to2V(P) . Thus a family of symmetric matrices is contained in a CJA if andonly if its matrices commute.4If M ∈ A and pb(A) = {Q1, . . . , Qk} we haveM =k∑j=1a jQ j = ∑j∈ϕ(M)a jQ j , (2.1)6with ϕ(M) = { j : a j 6= 0} . It is easy to see that the orthogonal projectionmatrix on the range space R(M) of M is8Q(M) = ∑j∈ϕ(M)Q j. (2.2)Namely if Q is an orthogonal projection matrix belonging to A , we10saw that Q = ∑ j∈ϕ(Q) Q j , since Q is the orthogonal projection matrixon R(Q), thus rank(Q) = ∑ j∈ϕ(Q) rank(Q j) and so, if rank(Q) = 1 ,12Q ∈ pb(A) . If 1n Jn = 1n 1n1>n ∈ Q we will have 1n Jn ∈ pb(A) and we willthen put Q1 = 1n Jn and say that A is regular. The core model14Yo =m∑i=1Xiβi +w∑i=m+1Xiβ˜i (2.3)is orthogonal, ORT, if matrices Mi = XiX>i , i = 1, . . . , w commute. Let16then {Q1, . . . , Qk} be the principal basis of a CJA containing M1, . . . , Mw .If we assume that the β1, . . . ,βm are fixed and that the β˜m+1, . . . , β˜w have18null mean vectors and variance-covariance matrices σ2m+1 Icm+1 , . . . ,σ2w Icw ,Yo will have mean vector20uo =m∑i=1Xiβi (2.4)and variance covariance matrices22Vo =w∑l=m+1σ2i Mi with Mi = XiX>i , i = 1, . . . , w . (2.5)Now24Mi =k∑j=1bi, jQ j , i = 1, . . . , w, (2.6)so26Vo =k∑j=1γ jQ j (2.7)818 S. S. FERREIRA ET AL.withγ j =w∑i=m+1bi, jσ2, j = 1, . . . , k. (2.8)2Moreover uo will spanΩo = R([X1 . . . Xm]) =( m∑i=1Mi). (2.9)4Now M = ∑mi=1 Mi and T = Q(M) will belong to A . We can reorder thematrices in pb(A) to have6T =h∑j=1Q j . (2.10)Since R(Mi) ⊆ R(T) , i = 1, . . . , m we will have bi, j = 0 , j = h+ 1, . . . , k ,8i = 1, . . . , m . Then the matrix B = [bi, j] will beB =[B1,1 0B2,1 B2,2](2.11)10with B1,1 a matrix of type m × h , B2,1 a matrix of type (w − m) × hand B2,2 a matrix of type (w − m) × (k − h) . 0 is a null matrix of type12m× (k− h) .Putting σ2 = (σ2m+1, . . . ,σ2w) , γ1 = (γ1, . . . ,γh) and γ2 =14(γh+1, . . . ,γk) we haveγ(1) = B>2,1σ2γ(2) = B>2,2σ2 .(2.12)16Then, if the row vectors of B2,2 are linearly independent the columnvectors of B>2,2 will be linearly independent and we have, see [7],18 σ2 = (B>2,2)>γ(2)γ(1) = B>2,1(B>2,2)>γ(2) (2.13)with C+ the Moore-Penrose inverse of C , see [8]. Thus the parameters of20γ(2) and σ2 of random effects part of the core model will determine eachother. Then the random effects part of the core model segregates itself as a22separate sub-model. We thus say that these core models are segregated.3. Estimation24The orthogonal projection matrices on Ω = R(L) will be Q(Ω) =LL+ . Let now the row vectors of K constitute an orthonormal basis26L ORTHOGONAL MODELS 819for the orthogonal complement Ω⊥ of Ω, then, with n the number ofcomponents of Y we will have2Q(Ω⊥) = In −Q(Ω) = K>K . (3.1)The orthogonal projections YΩ and Y⊥Ω of Y in Ω and Ω⊥ will be4YΩ = LYo + eΩ (3.2)and6Y⊥Ω = e⊥Ω , (3.3)thus, if we assume that cross-variance matrices8Σupslope(β˜i , e), i = m + 1, . . . , w, (3.4)we will have10Σupslope(YΩ, Y⊥Ω) = 0n×n (3.5)since12Σupslope(eΩ, e⊥Ω) = 0n×n . (3.6)Thus, taking S = ‖Y⊥Ω‖2 = ‖e⊥Ω‖2 we will have E(S) = gσ2 with14g = n − no where no = rank(L) . Moreover, since the column vectorsof L are linearly independent we have L+L = In, so that16Z = L+Y = L+LL+Y = L+YΩ = Yo + L+eΩ (3.7)= Yo + L+LL+e = Yo + L+e , (3.8)18with L+e independent from Yo, so Z will have mean vector uo andvariance covariance matrix20Vo +σ2L+L+>. (3.9)Let now the g j row vectors of A j , j = 1, . . . , k constitute an orthonormal22basis for R(Q j) , j = 1, . . . , k . Takingη j = A juo ,η˜ j = A jZ ,j = 1, . . . , k, (3.10)24since u spans ω and T = ∑hj=1 Q j is the orthogonal projection matrix onω , we will have η j = 0, j = h + 1, . . . , k . Moreover since the Q1, . . . , Qk26are pairwise orthogonal, the variance-covariance matrix of η˜ j will beγ j Ig j +σ2 A jL+L+>A>j , thus, taking28S j = ‖η˜ j‖2 , j = 1, . . . , k, (3.11)820 S. S. FERREIRA ET AL.we haveE(S j) = g jγ j + t jσ2 , j = m + 1, . . . , w, (3.12)2see [9], with t j = trace(A jL+L+>A>j ) , j = 1, . . . , k and g j = rank(Q j) ,j = 1, . . . , k . Thus we have the unbiased estimators4γ˜ j =S jg j− t jg jσ2, j = m + 1, . . . , k (3.13)for the components of γ(2) . If the core model are segregated we also have6the unbiased estimatorsσ˜2 = (B>2,2)>γ˜(2)γ˜(1) = B>2,1(B>2,2)>γ˜(2) .(3.14)8As to estimable vectors, putting{X = [X1 . . . Xm]β = [β1> . . .βm>]>(3.15)10we haveuo = Xβ (3.16)12and so the mean vector of Y will beu = Luo = LXβ . (3.17)14Given a matrix G, GY and GYω will have the same mean vector sinceboth Y and Yω have mean vector u . Now16ψ = Gβ (3.18)is estimable if there is a linear unbiased estimator ψo for ψ .18The mean vector of UY is ULXβ, and UY is an unbiased estimatorof ψ if20ULX = G, (3.19)this is if22ψ = Gβ = ULuo. (3.20)Since24uo = Tuo =h∑j=1Q juo =h∑j=1A>j A juo =h∑j=1A>j η j (3.21)L ORTHOGONAL MODELS 821we haveψ = ULh∑j=1A>j η j , (3.22)2thusψ∗ = ULh∑j=1A>j η˜ j (3.23)4will be an unbiased estimator of ψ .4. The orthogonal case6Let us assume that L+ = L> , then matricesQ j = LQ jL>, j = 1, . . . , w, (4.1)8will be pairwise orthogonal projection matrices which are orthogonal.Moreover10w∑j=1Q j = LL> = Q(Ω) (4.2)with Ω = R(L) . Thus adding to the Q j , j = 1, . . . , w the matrix12Q⊥ = In − LL> (4.3)we get the principal basis of a complete CJA, A .14If the row vectors of A j constitute an orthogonal basis for R(Q j) ,j = 1, . . . , w we have A>j A j = Q j and A j A>j = Ig j , with g j = rank(Q j)16= rank(Q j) , j = 1, . . . , w . Then, withA j = A jL>, j = 1, . . . , w (4.4)18we will haveQ j = A>j A jIg j = A j A>j ,j = 1, . . . , w, (4.5)20and the row vectors of A j will constitute an orthogonal basis for R(Q j) .Thus, since L+L = In,22 A ju = A jL>Luo = A juo = η jA jY = A jL>LZ = A jZ = η˜ j ,j = 1, . . . , w. (4.6)822 S. S. FERREIRA ET AL.Besides this the variance-covariance matrix of Y will beV = L( w∑j=1γ jQ j)L> +σ2 In =w∑j=1γ jQ j +σ2 In2=w∑j=1(γ j +σ2)Q j +σ2Q⊥ , (4.7)so4 V−1 = ∑wj=1(γ j +σ2)−1Q j +1σ2Q⊥det(V) = ∏wj=1(γ j +σ2)g j(σ2)g ,(4.8)with g = n− no . Thus6(y− u)>V−1(y− u) =w∑j=1(y− u)>A>A(y− u)(γ j +σ2)+y>Q⊥yσ2=w∑j=1‖η˜ j − η j‖2γ j +σ2+w∑j=m+1S jγ j +σ2+Sσ2, (4.9)8where, as before, S j = ‖η˜ j‖2 , j = m + 1, . . . , w, andS = y>Q⊥y = ‖yΩ⊥‖2. (4.10)10Thus when normality is assumed Y will have the densityn(y) =e− 12[m∑j=1‖η˜ j−η j‖2γ j+σ2 +w∑j=m+1S jγ j+σ2 +Sσ2](2pi)n2w∏j=1(γ j +σ2)g j2 σ g. (4.11)12We now haveProposition 1. The η˜ j , j = 1, . . . , m, the S j , j = m + 1, . . . , w and S will14be sufficient complete statistics.Proof. Applying the Factorization theorem we see that the statistics16are sufficient. Moreover they will be complete, see [5], since the normaldistribution belongs to the exponential family and, for these models,18the parameter space contains the cartesian product of non degenerateintervals. ¤20Corollary 2. The ψ˜ j, the γ˜(1) , γ˜(2) and σ˜2 will be UMVUE.Proof. The thesis follows from the proposition and the Blackwell-Lehman-22Scheffe´ theorem. ¤L ORTHOGONAL MODELS 823References[1] M. Fonseca, J.T. 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Birkes, Sufficient conditions for38orthogonal designs in mixed linear models, J. Statist. Plann. Inference,Vol. 73 (1998), pp. 373–389.40824 S. S. FERREIRA ET AL.[16] D. Van Leeuwen, D. Birkes and J. Seely, Balance and orthogonality indesigns for mixed classification models, Ann. Statist., Vol. 27 (1999) ,2pp. 1927–1947.Received April, 20094

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Inference for L orthogonal models

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