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Characterizations and dispersion-matrix robustness of efficiently estimable parametric functionals in linear models with nuisance parameters. (English) Zbl 0709.62063

A linear model containing main and nuisance parameters is considered. The effect of the presence of nuisance parameters and an alternative dispersion matrix on the precision of the best linear unbiased estimators of an arbitrary functional of the main parameters is investigated. Subspaces of those functionals for which the least squares estimators are robust against both criteria are obtained.
Reviewer: E.Ursianu

MSC:

62J99 Linear inference, regression
62H12 Estimation in multivariate analysis
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[1] Baksalary, J.K., A study of the equivalence between a Gauss-markoff model and its augmentation by nuisance parameters, Math. operationsforsch. statist. ser. statist., 15, 3-35, (1984) · Zbl 0556.62045
[2] Baksalary, J.K., Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors, (), 113-142
[3] Baksalary, J.K., A rank characterization of linear models with nuisance parameters and its application to block designs, J. statist. plann. inference, 22, 173-179, (1989) · Zbl 0669.62065
[4] Baksalary, J.K.; Kala, R.; Kłaczyński, K., The matrix inequality M ⩾ {\bfb}*{\bfmb}, Linear algebra appl., 54, 77-86, (1983) · Zbl 0516.15010
[5] Baksalary, J.K.; Nordström, K.; Styan, G.P.H., Löwner-ordering antitonicity of generalized inverses of Hermitian matrices, Linear algebra appl., (1989), to appear · Zbl 0697.15007
[6] Ehrenfeld, S., On the efficiency of experimental designs, Ann. math. statist., 26, 247-255, (1955) · Zbl 0064.38505
[7] Ehrenfeld, S., Complete class theorems in experimental design, (), 57-67
[8] Fellman, J., On the effect of “nuisance” parameters in linear models, Sankhyā ser. A, 38, 197-200, (1976) · Zbl 0387.62060
[9] Fellman, J., Estimation in linear models with nuisance parameters, Statist. decisions, Suppl. No. 2, 161-164, (1985)
[10] Hartwig, R.E., How to partially order regular elements, Math. japonica, 25, 1-13, (1980) · Zbl 0442.06006
[11] Hartwig, R.E.; Styan, G.P.H., Partially ordered idempotent matrices, (), 361-383
[12] Kiefer, J., Optimum experimental designs (with discussion), J. roy. statist. soc. ser. B, 21, 272-319, (1959) · Zbl 0108.15303
[13] Kubáček, L., Elimination of nuisance parameters in a regression model, Math. slovaca, 36, 137-144, (1986) · Zbl 0605.62081
[14] Löwner, K., Über monotone matrixfunktionen, Math. Z., 38, 177-216, (1934) · JFM 60.0055.01
[15] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear and multilinear algebra, 2, 269-292, (1974)
[16] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic Orlando · Zbl 0437.26007
[17] Mitra, S.K.; Rao, C.R., Some results in estimation and tests of linear hypotheses under the Gauss-markoff model, Sankhyā ser. A, 30, 281-290, (1968) · Zbl 0197.15803
[18] Mitra, S.K.; Rao, C.R., Projections under seminorms and generalized Moore-Penrose inverses, Linear algebra appl., 9, 155-167, (1974) · Zbl 0296.15002
[19] Nordström, K.; Fellman, J., Characterizations of efficiently estimable parametric functions in linear models with nuisance parameters, () · Zbl 0709.62063
[20] Nordström, K.; von Rosen, D., Algebra of subspaces with applications to problems in statistics, (), 603-614
[21] Pringle, R.M.; Rayner, A., Generalized inverse matrices with applications to statistics, (1971), Griffin London · Zbl 0231.15008
[22] Puntanen, S.; Styan, G.P.H., On the equality of the ordinary least squares estimator and the best linearunbiased estimator (with comments), Amer. statist., 43, 153-164, (1989)
[23] Rao, C.R., On the linear combination of observations and the general theory of least squares, Sankhyā, 7, 237-256, (1946) · Zbl 0063.06423
[24] Rao, C.R., Least squares theory using an estimated dispersion matrix and its application to measurement of signals, (), 355-372 · Zbl 0189.18503
[25] Rao, C.R., A note on a previous lemma in the theory of least squares and some further results, Sankhyā ser. A, 30, 259-266, (1968) · Zbl 0197.15802
[26] Rao, C.R., Linear statistical inference and its applications, (1973), Wiley New York · Zbl 0169.21302
[27] Rao, C.R., Representations of best linear unbiased estimators in the Gauss-markoff model with a singular dispersion matrix, J. multivariate anal., 3, 276-292, (1973) · Zbl 0276.62068
[28] Rao, C.R., Projectors, generalized inverses and the BLUE’s, J. roy. statist. soc. ser. B, 36, 442-448, (1974) · Zbl 0291.62077
[29] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York
[30] 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
[31] Rosenberg, M., Range decomposition and generalized inverse of nonnegative Hermitian matrices, SIAM rev., 11, 568-571, (1969) · Zbl 0248.15015
[32] Seber, G.A.F., Linear regression analysis, (1977), Wiley New York · Zbl 0354.62055
[33] Shinozaki, N.; Sibuya, M., Product of projectors, (1974), Scientific Center, IBM Japan Tokyo, Research Report
[34] Stein, R.A., Linear model estimation, projection operators, and conditional inverses, ()
[35] Stepniak, C.; Wang, S.-G.; Wu, C.F.J., Comparison of linear experiments with known covariances, Ann. statist., 12, 358-365, (1984) · Zbl 0546.62004
[36] Styan, G.P.H., Schur complements and linear statistical models, (), 37-75
[37] Zyskind, G., On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models, Ann. math. statist., 38, 1092-1109, (1967) · Zbl 0171.17103
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