Abstract
Minimax nonhomogeneous linear estimators of scalar linear parameter functions are studied in the paper under restrictions on the parameters and variance-covariance matrix. The variance-covariance matrix of the linear model under consideration is assumed to be unknown but from a specific set R of nonnegativedefinite matrices. It is shown under this assumption that, without any restriction on the parameters, minimax estimators correspond to the least-squares estimators of the parameter functions for the “worst” variance-covariance matrix. Then the minimax mean-square error of the estimator is derived using the Bayes approach, and finally the exact formulas are derived for the calculation of minimax estimators under elliptical restrictions on the parameter space and for two special classes of possible variance-covariance matrices R. For example, it is shown that a special choice of a constant q 0 and a matrixW 0 defining one of the above classes R leads to the well known Kuks—Olman admissible estimator (see [16]) with a known variance-covariance matrixW 0. Bibliography:32 titles.
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References
P. Alson, “Centered ellipsoids for which an admissible linear estimation is the minimax estimator,”Statistics,24, 85–94 (1993).
J. K. Baksalary and T. Mathew, “Admissible linear estimation in general Gauss-Markov model with an incorrectly specified dispersion matrix,”J. Multivariate Anal.,27, 53–67 (1988).
J. K. Baksalary and A. Markiewich, “Admissible linear estimators of arbitrary vectors of parameter functions in the general Causs-Markov model,”J. Statist. Planning Inference,26, 161–171 (1990).
D. Birkes and Y. Dodge,Alternative Methods of Regressions, John Wiley, New York (1993).
L. D. Broemeling,Bayesian Analysis of Linear Models, Marcel Decker, New York (1985).
H. Bunke and O. Bunke,Statistical Inference in Linear Models, Vol. 1, John Wiley, Chichester (1986).
S. L. Dixon and J. W. McKean, “Rank-based analysis of the heteroscedastic linear model,”J. Amer. Statist. Assoc.,91, 699–712 (1996).
N. Gaffke and B. Heilinger, “Admissible and minimax linear estimators in linear models with restricted parameter space,”Statistics 20, 487–508 (1989).
W. Hardle,Applied nonparametric regression, Cambridge University Press, Cambridge (1990).
B. heilinger, “Linear Bayes estimation in linear models with partially restricted parameter space,”J. Statist. Planning Inference,36, 175–184 (1993).
K. Hoffman, “All admissible linear estimators of the regression parameter vector in the class of an arbitrary parameter subset,”J. Statist. Planning Inference,48, 371–377 (1995).
R. A. Horn and C. A. Johnson,Matrix Analysis, Cambridge University Press, Cambridge (1985).
T. Honda, “Minimax estimators in the MANOVA model for quadratic loss and unknown covariance matrix,”J. Multivar. Anal.,36, 113–120 (1991).
J. Kleffe, “A minimax property of translation invariant estimators for variance components,”Zastosowania Matematyki Applicationes Mathematicae,XVI, No. 3, 445–458 (1979).
W. Klonecki and S. Zontek, “On the structure of admissible linear estimators,”J. Multivar. Anal.,24, 11–30 (1988).
J. Kuks and V. Olman, “Minimax linear estimators of regression coefficients,”Izv. Akad. Nauk Eston. SSR,21, 66–72 (1972).
J. Kuks, “Minimax estimation of regression coefficients,”Izv. Akad. Nauk Eston. SSR,21, 73–78 (1972).
H. Lauter, “A minimax linear estimator for linear parameters under restrictions in the form of inequalities,”Math. Operationsforsch. und Statist.,5, 689–695 (1975).
D. Lindley and A. F. M. Smith, “Bayes estimators for the linear models,”J. Royal Statistical Soc.,B33, 1–41 (1972).
C. Y. Lu and W. X. Li, “Admissibility of linear estimators in linear models with incomplete ellipsoidal constraints,”Acta Math. Sinica,37, 289–295 (1994).
D. W. Marquard and R. D. Snee, “Ridge regression in practice,”American Statistician,29, 3–20 (1975).
H. G. Muller,Nonparametric regression analysis of longitudinal data, Lecture Notes in Statistics, Vol. 46, Springer-Verlag, Berlin-Heidelberg (1988).
O. Nakonechny,Minimax Estimation of Functionals of Solutions of Variation Equations in Hilbert Spaces [in Russian], KGU, Kiev (1985).
C. R. Rao,Linear Statistical Inference and Its Applications, John Wiley, New York (1973).
C. R. Rao, “Estimation of parameters in linear models,”Ann. Stat.,10, 245–255 (1976).
R. T. Rockafeller,Convex Analysis, Princeton University Press, Princeton (1970).
A. Schick, “Efficient estimates in linear and nonlinear regression with heteroscedastic errors,”J. Statist. Planning Inference,58, 371–387 (1997).
G. Smith and F. Campbell, “A critique of some regression methods,”J. American Statistical Associations,75, 87–103 (1980).
R. G. Staudte and R. G. Sheather,Robust Estimation and Testing, John Wiley, New York (1990).
Q. L. Wu, “Inadmissibility and admissibility results for unbiased loss estimates based on Gauss-Markov estimators,”Acta Math. Appl. Sinica (English Ser.),9, 281–288 (1993).
J. L. Zhan, “Admissible linear estimators of regression coefficients under quadratic loss,”Chinese J. Math.,22, 53–63 (1994).
S. Zontek, “On characterization of linear admissible estimators: An extension of a result due to C. R. Rao,”J. Multivariate Analysis,36, 1–12 (1987).
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Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 79–92.
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Michálek, J., Nakonechny, O. Minimax estimates of a linear parameter function in a regression model under restrictions on the parameters and variance-covariance matrix. J Math Sci 102, 3790–3802 (2000). https://doi.org/10.1007/BF02680236
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DOI: https://doi.org/10.1007/BF02680236