# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Some properties of projectors associated with the WLSE under a general linear model. (English) Zbl 1141.62043
Summary: Projectors associated with a particular estimator in a general linear model play an important role in characterizing statistical properties of the estimator. A variety of new properties were derived on projectors associated with the weighted least-squares estimator (WLSE). These properties include maximal and minimal possible ranks, rank invariance, uniqueness, idempotency, and other equalities involving the projectors. Applications of these properties were also suggested. The proofs of the main theorems demonstrate how to use the matrix rank method for deriving various equalities involving the projectors under the general linear model.

##### MSC:
 62H12 Multivariate estimation 62J05 Linear regression 15A09 Matrix inversion, generalized inverses
Full Text:
##### References:
 [1] Baksalary, J. K.; Kala, R.: Two relations between oblique and $\Lambda$-orthogonal projectors, Linear algebra appl. 24, 99-103 (1979) · Zbl 0401.15004 · doi:10.1016/0024-3795(79)90150-2 [2] Baksalary, J. K.; Puntanen, S.: Weighted-least-squares estimation in the general Gauss -- Markov model, Statistical data analysis and inference, 355-368 (1989) · Zbl 0736.62045 [3] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses: theory and applications, (2003) · Zbl 1026.15004 [4] Campbell, S. L.; Meyer, C. D.: Generalized inverses of linear transformations, corrected reprint of the 1979 original, (1991) · Zbl 0732.15003 [5] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear and multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003 [6] Mitra, S. K.; Moore, B. J.: Gauss -- Markov estimation with an incorrect dispersion matrix, Sankhyā ser. A 35, 139-152 (1973) · Zbl 0277.62044 [7] Mitra, S. K.; Rao, C. R.: Projections under seminorms and generalized Moore -- Penrose inverses, Linear algebra appl. 9, 155-167 (1974) · Zbl 0296.15002 · doi:10.1016/0024-3795(74)90034-2 [8] Penrose, R.: A generalized inverse for matrices, Proc. Cambridge philos. Soc. 51, 406-413 (1955) · Zbl 0065.24603 [9] Rao, C. R.: Projectors, generalized inverses and the blue’s, J. roy. Statist. soc. Ser. B 36, 442-448 (1974) · Zbl 0291.62077 [10] Rao, C. R.; Mitra, S. K.: Further contributions to the theory of generalized inverse of matrices and its applications, Sankhyā ser. A 33, 289-300 (1971) · Zbl 0236.15005 [11] Rao, C. R.; Mitra, S. K.: Generalized inverse of matrices and its applications, (1971) · Zbl 0236.15004 [12] Rao, C. R.; Yanai, H.: General definition and decomposition of projectors and some applications to statistical problems, J. statist. Plann. inference 3, 1-17 (1979) · Zbl 0427.62046 · doi:10.1016/0378-3758(79)90038-7 [13] Takane, Y.; Yanai, H.: On oblique projectors, Linear algebra appl. 289, 297-310 (1999) · Zbl 0930.15003 · doi:10.1016/S0024-3795(98)10180-5 [14] Y. Takane, Y. Tian, H. Yanai, On constrained generalized inverses of matrices and their properties, Ann. Inst. Math. Statist., in press. · Zbl 1133.62339 · doi:10.1007/s10463-006-0075-3 [15] Tian, Y.: The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast asian bull. Math. 25, 745-755 (2002) · Zbl 1007.15005 · doi:10.1007/s100120200015 [16] Tian, Y.; Cheng, S.: The maximal and minimal ranks of A-BXC with applications, New York J. Math. 9, 345-362 (2003) · Zbl 1036.15004 · emis:journals/NYJM/j/2003/9-18nf.htm [17] Yanai, H.: Some generalized forms of least squares g-inverse, minimum norm g-inverse, and Moore -- Penrose inverse matrices, Comput. statist. Data anal. 10, 251-260 (1990) · Zbl 0825.62550 · doi:10.1016/0167-9473(90)90005-3