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2016 | 4 | 1 | 130-140
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Matrix rank/inertia formulas for least-squares solutions with statistical applications

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Least-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z) subject to the LSS of AZ = B, and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
1
Strony
130-140
Opis fizyczny
Daty
otrzymano
2015-06-11
zaakceptowano
2016-02-01
online
2016-02-12
Twórcy
autor
  • China Economics and Management Academy, Central University of Finance and Economics, Beijing, China
autor
  • College of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, China
Bibliografia
  • [1] B.E. Cain, E.M. De Sá. The inertia of Hermitian matrices with a prescribed 2 × 2 block decomposition. Linear Multilinear Algebra 31(1992), 119–130. [Crossref]
  • [2] Y. Chabrillac, J.P. Crouzeix. Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra Appl. 63(1984), 283–292.
  • [3] C.M. da Fonseca. The inertia of certain Hermitian block matrices. Linear Algebra Appl. 274(1998), 193–210.
  • [4] J. Dancis. The possible inertias for a Hermitianmatrix and its principal submatrices. Linear Algebra Appl. 85(1987), 121–151.
  • [5] J. Dancis. Several consequences of an inertia theorem. Linear Algebra Appl. 136(1990), 43–61.
  • [6] F.A. Graybill. An Introduction to Linear Statistical Models. Vol. I, McGraw–Hill, 1961.
  • [7] L. Guttman. General theory and methods for matric factoring. Psychometrika 9(1944), 1–16. [Crossref]
  • [8] A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.
  • [9] E.V. Haynsworth. Determination of the inertia of a partitioned Hermitian matrix. Linear Algebra Appl. 1(1968), 73–81.
  • [10] E.V. Haynsworth, A.M. Ostrowski. On the inertia of some classes of partitionedmatrices. Linear Algebra Appl. 1(1968), 299– 316.
  • [11] I.J. James. The topology of Stiefel manifolds. Cambridge University Press, 1976.
  • [12] C.R. Johnson, L. Rodman. Inertia possibilities for completions of partial Hermitian matrices. Linear Multilinear Algebra 16(1984), 179–195. [Crossref]
  • [13] Y. Liu, Y. Tian. More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications. Numer. Linear Algebra Appl. 15(2008), 307–325. [WoS]
  • [14] Y. Liu, Y. Tian. Max-min problems on the ranks and inertias of the matrix expressions A − BXC ± (BXC)* with applications. J. Optim. Theory Appl. 148(2011), 593–622. [WoS]
  • [15] G. Marsaglia, G.P.H. Styan. Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2(1974), 269–292. [Crossref]
  • [16] S.K. Mitra. The matrix equations AX = C, XB = D. Linear Algebra Appl. 59(1984), 171–181.
  • [17] R. Penrose. A generalized inverse for matrices. Proc. Cambridge Phil. Soc. 51(1955), 406–413.
  • [18] S. Puntanen, G.P.H. Styan, J. Isotalo. Matrix Tricks for Linear Statistical Models, Our Personal Top Twenty. Springer, 2011.
  • [19] S.R. Searle. Linear Models. Wiley, 1971.
  • [20] Y. Tian. Equalities and inequalities for inertias of Hermitian matrices with applications. Linear Algebra Appl. 433(2010), 263–296. [WoS]
  • [21] Y. Tian. Maximization and minimization of the rank and inertia of the Hermitian matrix expression A − BX − (BX)* with applications. Linear Algebra Appl. 434(2011), 2109–2139. [WoS]
  • [22] Y. Tian. Solving optimization problems on ranks and inertias of some constrained nonlinearmatrix functions via an algebraic linearization method. Nonlinear Analysis 75(2012), 717–734.
  • [23] Y. Tian. Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications. Linear Algebra Appl. 437(2012), 835–859. [WoS]
  • [24] Y. Tian. Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A + BXB*. Math. Comput. Modelling 55(2012), 955–968.
  • [25] Y. Tian. Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions. Banach J. Math. Anal. 8(2014), 148–178.
  • [26] Y. Tian. A new derivation of BLUPs under a random-effects model. Metrika 78(2015), 905–918. [Crossref][WoS]
  • [27] Y. Tian. A matrix handling of predictions under a general linear random-effects model with new observations. Electron. J. Linear Algebra 29(2015), 30–45. [Crossref]
  • [28] Y. Tian. A survey on rank and inertia optimization problems of the matrix-valued function A + BXB*. Numer. Algebra Contr. Optim. 5(2015), 289–326.
  • [29] Y. Tian. How to characterize properties of general Hermitian quadratic matrix-valued functions by rank and inertia. In: Advances in Linear Algebra Researches, I. Kyrchei, (ed.), Nova Publishers, New York, pp. 150–183, 2015.
  • [30] Y. Tian. Some equalities and inequalities for covariance matrices of estimators under linear model. Statist. Papers, DOI:10.1007/s00362-015-0707-x. [Crossref]
  • [31] Y. Tian,W. Guo. On comparison of dispersionmatrices of estimators under a constrained linear model. Stat. Methods Appl., DOI:10.1007/s10260-016-0350-2. [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_spma-2016-0013
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