Estimation of subparameters by IPM method. IPM yöntemi ile alt parametrelerin tahmini

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1 SAÜ Fen Bil Der 0. Cilt. Sayı s Estimation of subparameters by IPM method Melek Eriş * * Nesrin Güler ABSTRACT Geliş/Received Kabul/Accepted In this study a general partitioned linear model y X V y X X V is considered to determine the best linear unbiased estimators (BLUEs) of subparameters X X. Some results are given related to the BLUEs of subparameters by using the inverse partitioned matrix (IPM) method based on a generalized inverse of a symmetric block partitioned matrix which is obtained from the fundamental BLUE equation. Keywords: BLUE generalized inverse general partitioned linear model ÖZ IPM yöntemi ile alt parametrelerin tahmini Bu çalışmada X ve X alt parametrelerinin en iyi lineer yansız tahmin edicilerini (BLUE larını) belirlemek için bir y X V y X X V genel parçalanmış lineer modeli ele alınmıştır. Temel BLUE denkleminden elde edilen simetrik blok parçalanmış matrisin bir genelleştirilmiş tersine dayanan parçalanmış matris tersi (IPM) yöntemi kullanılarak alt parametrelerin BLUE ları ile ilgili bazı sonuçlar verilmiştir. Anahtar Kelimeler: BLUE genelleştirilmiş ters genel parçalanmış lineer model Sorumlu Yazar / Corresponding Author Karadeniz Teknik Üniversitesi Fen Fakültesi İstatistik ve Bilgisayar Bilimleri Bölümü Trabzon - melekeris@ktu.edu.tr Sakarya Üniversitesi Fen Edebiyat Fakültesi İstatistik Bölümü Sakarya - nesring@sakarya.edu.tr

2 M. Eriş N. Güler Estimation of subparameters by IPM method. INTRODUCTION Consider the general partitioned linear model y X X X () where nx y R is an observable rom vector X X : X R is a known matrix with X R X R px : R is a p x vector of unknown parameters with R p x nx R R is a rom error vector. Further the expectation E y n X the covariance matrix Cov y V R is a known nonnegative definite matrix. We may denoted the model () as a triplet y X V y X X V. () Partitioned linear models are used in the estimations of partial parameters in regression models as well as in the investigations of some submodels reduced models associated with the original model. In this study we consider the general partitioned linear model we deal with the best linear unbiased estimators (BLUEs) of subparameters under this model. Our main purpose is to obtain the BLUEs of subparameters X X under by using the inverse partitioned matrix (IPM) method which is introduced by Rao [] for statistical inference in general linear models. We also investigate some consequences on the BLUEs of subparameters obtained by using IPM approach. Under the linear models BLUE has been investigated by many statisticians. Some valuable properties of BLUE have been obtained e.g. [-6]. By applying matrix rank method some characterizations of BLUE have been given by Tian [78]. IPM method for the general linear model with linear restrictions has been considered by Baksalary [9]. The BLUE of. PRELIMINARIES X under denoted as BLUE X is defined to be an unbiased linear estimator Gy such that its covariance matrix Cov Gy is minimal in the Löwner sense among all covariance matrices Cov Fy such that Fy is unbiased for X. It is well-known see e.g. [0] that Gy BLUE X if only if G satisfies the fundamental BLUE equation X G X : VQ : 0 () where Q Px with x P is orthogonal projector onto the column space C X. Note that the equation () has a unique solution if only if rank X : V n the observed value of Gy is unique with probability if only if is consistent i.e. y C X : V C X : VQ holds with probability ; see []. In the study it is assumed that the model is consistent. The corresponding condition for Ay to be BLUE of an estimable parametric function K is : K : 0 A X VQ K is estimable under C X. Recall that a parametric function if only if C K in particular X X is estimable under if only if C X C X ; see []. 0 The fundamental BLUE equation given in () equivalently expressed as follows. Gy BLUE X if only if there exists a matrix p L R such that G is solution to V X G 0 i.e X 0 L X G 0 Z. () L X Partitioned matrices their generalized inverses play an important role in the concept of linear models. According to Rao [] the problem of inference from a linear model can be completely solved when one has obtained an arbitrary generalized inverse of the partitioned matrix Z. This approach based on the numerical evaluation of an inverse of the partitioned matrix Z is known as the IPM method see [-5]. C C Let the matrix C be an arbitrary C C generalized inverse of Z i.e. C is any matrix satisfying the equation ZCZ Z where C R C R () is n. Then one solution to the (consistent) equation 60 SAÜ Fen Bil Der 0. Cilt. Sayı s

3 Estimation of subparameters by IPM method M. Eriş N. Güler G C C 0 C X. (5) L C C X C X Therefore we see that BLUE X XCy XC y which is one representation for the BLUE of X under. If we let C vary through all generalized inverses of Z we obtain all solutions to () thereby all representations Gy for the BLUE of X under. As further reference for submatrices C i i their statistical applications see [6-].. SOME RESULTS ON A GENERALIZED INVERSE OF Z Some explicit algebraic expression for the submatrices of C was obtained in [5 Theorem.]. The purpose of this section is to extend this theorem to x symmetric block partitioned matrix to obtain the BLUEs of subparameters their properties. Let D Z expressed as D0 D D V X X D E F F X 0 0 E F F X 0 0 n where D0 R D R D R E E F F F F are conformable matrices Z sts for the set of all generalized inverse of Z. In the following theorem we collect some properties related to the submatrices of D given in (6). (6) Theorem. Let V X X D 0 D D E E F F F F be defined as before let C X C X. Then the following hold: 0 V X X D0 E E (i) X 0 0 D F F X 0 0 D F F is another choice of a generalized inverse. (ii) VD0 X XE X X E X X X D0 X 0 X D0 X 0. (iii) VD0 X XE X X E X X X D0 X 0 X D0 X 0. (iv) VD0V XEV X EV V X D0V 0 X D0V 0. VD X X F X X F X XD X X (v) XD X 0. VD X X F X X F X X D X X (vi) X D X 0. Proof: The result (i) is proved by taking transposes of either side of (6). We observe that the equations Va Xb X c Xd X a 0 X a 0 (7) are solvable for any d in which case a D0 Xd b E Xd c E Xd is a solution. Substituting this solution in (7) omitting d we have (ii). To prove (iii) we can write the equations Va Xb X c X d X a 0 X a 0 (8) which are solvable for any d. Then a D0 X d b E X d c E X d is a solution. Substituting this solution in (8) omitting d we have (iii). To prove (iv) the equations which are solvable for any d Va Xb X c Vd X a 0 X a 0 (9) are considered. In this case one solution is a D0Vd b EVd c EVd. If we substitute this solution in (9) omit d we have (iv). In view of the assumption C X C X we can consider the equations 0 Va Xb X c 0 X a X d X a 0 (0) Va Xb X c 0 X a 0 X a X d () for the proof of (v) (vi) respectively see [8 Theorem 7..8]. In this case a D X d b F X d c F X d is a solution for (0) a D X d b F X d c F X d is a solution for (). Substituting these solutions into corresponding equations omitting d d we obtain the required results. SAÜ Fen Bil Der 0. Cilt. Sayı s

4 M. Eriş N. Güler Estimation of subparameters by IPM method. IPM METHOD FOR SUBPARAMETERS The fundamental BLUE equation given in () can be accordingly written for Ay being the BLUE of estimable K that is Ay BLUE K exists a matrix if only if there p L R such that A is solution to A 0 Z. () L K Now assumed that X X are estimable under. If we take K X : 0 K 0 : X respectively from equation () we get the BLUE equations of subparameters X X. There exist p p p p L R L R L R L R such that G G are solution to following the equations respectively G y BLUE X V X X G 0 X 0 0 L X X 0 0 L 0 G y BLUE X X 0 0 L X () V X X G 0 X 0 0 L 0. () G D0 D D 0 L E F F X L E F F 0 D0 D D V X X E F F X 0 0 U E F F X 0 0 thereby we get G y X Dy U VD X D X D y 0 U X D y U X D y 0 0. Here y can be written as y XL X L VQL for some L L L since the model is assumed to be consistent. From Theorem we see that U VD X D X D X L X L VQL 0 0 U X D X L X L VQL 0 0 U X D y U X D X L X L VQL Moreover according to Theorem (i) we can replace D by E. Therefore BLUE X X Dy X E y is obtained. BLUE X X D y X E y is obtained by similar way The following results are easily obtained from Theorem (v) (vi) under. Therefore the following theorem can be given to determine the BLUE of subparameters by the IPM method. Theorem. Consider the general partitioned linear model the matrix D given in (6). Suppose that C X C X. Then 0 BLUE X X Dy X E y BLUE X X D y X E y. (5) Proof: The general solution of the matrix equation given in () is E BLUE X E BLUE X Cov BLUE X X X Cov BLUE X Cov BLUE X X F X X F X BLUE X X F X X FX. (6) (7) (8) 6 SAÜ Fen Bil Der 0. Cilt. Sayı s

5 Estimation of subparameters by IPM method M. Eriş N. Güler 5. ADDITIVE DECOMPOSITION OF THE BLUES OF SUBPARAMETERS The purpose of this section is to give some additive properties of BLUEs of subparameters under. Theorem. Consider the model assume that X X are estimable under. BLUE X BLUE X BLUE for X under. is always Proof: Let BLUE X BLUE X be given as in (5). Then we can write BLUE X BLUE X X D X D y. According to fundamental BLUE equation from Theorem (v) (vi) we see that X D X D X : X : VQ X : X : 0 for all y C X : VQ obtained.. Therefore the required result is The following results are easily obtained from Theorem (iv) (6)-(8). BLUE X E BLUE X BLUE X X Cov BLUE X X F X X F X X F X X F X. REFERENCES [] C. R. Rao Unified theory of linear estimation Sankhyā Ser. A : [Corrigendum (97) p. 9 p. 77.]. [] M. Nurhanen S. Puntanen Effect of deleting an observation on the equality of the OLSE BLUE Linear Algebra its Applications vol. 76 pp [] H. J. Werner C. Yapar Some equalities for estimations of partial coefficients under a general linear regression model Linear Algebra its Applications 7/ [] P. Bhimasankaram R. Saharay On a partitioned linear model some associated reduced models Linear Algebra its Applications vol. 6 pp [5] J. Gross S. Puntanen Estimation under a general partitioned linear model Linear Algebra its Applications vol. pp [6] B. Zhang B. Liu C. Lu A study of the equivalence of the BLUEs between a partitioned singular linear model its reduced singular linear models Acta Mathematica Sinica English Series vol. 0 no. pp [7] Y. Tian S. Puntanen On the equivalence of estimations under a general linear model its transformed models Linear Algebra its Applications vol. 0 pp [8] Y. Tian On properties of BLUEs under general linear regression models Journal of Statistical Planning Inference vol. pp [9] J. K. Baksalary P. R. Pordzik Inverse- Partitioned-Matrix method for the general Gauss- Markov model with linear restrictions Journal of Statistical Planning Inference vol. pp [0] C. R. Rao Least squares theory using an estimated dispersion matrix its application to measurement of signals. In: Le Cam L. M. Neyman J. eds. Proc. Fifth Berkeley Symposium on Mathematical Statistics Probability: Berkeley California 965/966. Vol.. Univ. Of California Press Berkeley [] G. Zyskind On canonical forms non-negative covariance matrices best simple least squares linear estimators in linear models The Annals of Mathematical Statistics vol. 8 pp [] C. R. Rao Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix Journal of Multivariate Analysis vol. pp [] I. S. Alalouf G. P. H. Styan Characterizations of estimability in the general linear model The Annals of Statistics vol. 7 pp [] S. Puntanen G.P.H. Styan J. Isotalo Matrix tricks for linear statistical models Our Personal Top Twenty Springer Heidelberg 0. [5] C. R. Rao A note on the IPM method in the unified theory of linear estimation Sankhyā Ser.A. vol. pp [6] H. Drygas A note on the Inverse-Partitioned- Matrix method in linear regression analysis Linear Algebra its Applications vol. 67 pp [7] F. J. Hall C. D. Meyer Generalized inverses of the fundamental bordered matrix used in linear SAÜ Fen Bil Der 0. Cilt. Sayı s

6 M. Eriş N. Güler Estimation of subparameters by IPM method estimation Sankhyā Ser. A. vol. 7 pp [Corrigendum (978) 0 p. 99.] [8] D. A. Harville Matrix algebra from a statisticians perspective Springer New York 997. [9] J. Isotalo Puntanen S. Styan G. P. H. A useful matrix decomposition its statistical applications in linear regression Commun. Statist. Theor. Meth. vol. 7 pp [0] R. M. Pringle A. A. Rayner Generalized Inverse Matrices with Applications to Statistics Hafner Publishing Company New York 97. [] C. R. Rao Some recent results in linear estimation Sankhyā Ser. B vol. pp [] C. R. Rao Linear Statistical Inference its Applications nd Ed. Wiley New York 97. [] H. J. Werner C. R. Rao s IPM method: a geometric approach. New Perspectives in Theoretical Applied Statistics (M. L. Puri J. P. Vilaplana & W. Wertz eds.). Wiley New York SAÜ Fen Bil Der 0. Cilt. Sayı s