×

Minimax linear regression estimation with symmetric parameter restrictions. (English) Zbl 0602.62054

The author considers a linear regression model \(E(y)=X\cdot \theta\), \(\theta\in \Theta\), cov y\(=W\) (W assumed to be positive definite) with restricted parameter space \(\Theta \subset R^ K\), where \(\Theta\) is compact and symmetric about some centre point \(\theta_ 0\in R^ K\). This includes, as special cases, ellipsoid constraints \((\theta - \theta_ 0)'H(\theta -\theta_ 0)\leq 1\) with a positive definite matrix H and linear constraints. The explicit form of the minimax linear estimator involves the second moment matrix of a least favourable prior distribution on \(\Theta\). Explicit minimax linear estimators are determined for the important special cases of the above inequality.
Reviewer: J.Kaufmann

MSC:

62J05 Linear regression; mixed models
62F15 Bayesian inference
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bandemer, H., (Theorie und Anwendung der Optimalen Versuchsplanung, Vol. I (1977), Akademie-Verlag: Akademie-Verlag Berlin) · Zbl 0391.62001
[2] Bandemer, H.; Pilz, J.; Fellenberg, B., Integral geometric prior distributions for Bayesian linear regression with bounded response, Statistics (1986), to appear · Zbl 0606.62071
[3] Berger, J. O., A robust generalized Bayes estimator and confidence region for a multivariate normal mean, Ann. Statist., 8, 716-761 (1980) · Zbl 0464.62026
[4] Berger, J. O., Selecting a minimax estimator of a multivariate normal mean, Ann. Statist., 10, 81-92 (1982) · Zbl 0485.62046
[5] Bock, M. E., Employing vague inequality information in the estimation of normal mean vectors (Estimators that shrink to closed convex polyhedra), (Gupta, S. S.; Berger, J. O., Statistical Decision Theory and Related Topics III, Vol. 1 (1982), Academic Press: Academic Press New York-London), 169-193 · Zbl 0598.62063
[6] Böhning, D., On an assumption of Bandemer and Ketzel in optimal experimental design theory, Math. Operationsforsch. Statist. Ser. Statist., 12, 497-502 (1981) · Zbl 0525.62066
[7] Bunke, O., Minimax linear, ridge and shrunken estimators for linear parameters, Math. Operationsforsch. Statist., 6, 697-701 (1975) · Zbl 0318.62051
[8] Chaloner, K., Optimal Bayesian experimental design for linear models, (Carnegie-Mellon Technical Report No. 238. Carnegie-Mellon Technical Report No. 238, Ph.D. Thesis (1982)) · Zbl 0554.62063
[9] Chaloner, K., Optimal Bayesian experimental design for linear models, Ann. Statist., 12, 283-300 (1984) · Zbl 0554.62063
[10] Chipman, J. S.; Rao, M. M., The treatment of linear restrictions in regression analysis, Econometrica, 32, 198-209 (1964) · Zbl 0143.43305
[11] Davis, W. W., Bayesian analysis of the linear model subject to linear inequality constraints, J. Amer. Statist. Assoc., 73, 573-579 (1978) · Zbl 0403.62046
[12] Ehrenfeld, S., Complete class theorems in experimental designs, (Proc. Third Berkeley Symp. 1 (1956), Univ. California Press: Univ. California Press Berkeley-Los Angeles), 57-67 · Zbl 0075.14703
[13] Escobar, L. A.; Skarpness, B., A closed form solution for the least squares regression problem with linear inequality constraints, Commun. Statist. — Theory Methods, 13, 9, 1127-1134 (1984) · Zbl 0552.62054
[14] Gladitz, J.; Pilz, J., Construction of optimal designs in random coefficient regression models, Math. Operationsforsch. Statist. Ser. Statist., 13, 371-385 (1982) · Zbl 0498.62061
[15] Gladitz, J.; Pilz, J., Bayes designs for multiple linear regression on the unit sphere, Math. Operationsforsch. Statist. Ser. Statist., 13, 491-506 (1982) · Zbl 0515.62071
[16] Goldstein, M., General variance modifications for linear Bayes estimators, J. Amer. Statist. Assoc., 78, 383, 616-618 (1983) · Zbl 0534.62015
[17] Hoffmann, K., Admissibility of linear estimators with respect to restricted parameter sets, Math. Operationsforsch. Statist. Ser. Statist., 8, 425-438 (1977) · Zbl 0387.62007
[18] Hoffmann, K., Characterization of minimax linear estimators in linear regression, Math. Operationsforsch. Statist. Ser. Statist., 10, 19-26 (1979) · Zbl 0417.62051
[19] Hoffmann, K., Admissible improvements of the least squares estimator, Math. Operationsforsch. Statist. Ser. Statist., 11, 373-388 (1980) · Zbl 0456.62056
[20] Humak, K. M.S., (Statistische Verfahren zur Modellbildung, Vol. I (1977), Akademie-Verlag: Akademie-Verlag Berlin) · Zbl 0353.62001
[21] Ito, T., Methods of estimation for multi-market disequilibrium models, Econometrica, 48, 97-125 (1980) · Zbl 0427.90030
[22] Kraft, O., A maximum linear estimator for linear parameters under restrictions in form of inequalities, Statistics (1984), submitted
[23] Kuks, J.; Olman, V., Minimax linear estimation of regression coefficients (Russian), Izvestija Akademii Nauk Estonskoi SSR, 20, 480-482 (1971) · Zbl 0282.62054
[24] Kuks, J.; Olman, V., Minimax linear estimation of regression coefficients II (Russian), Izvestija Akademii Nauk Estonskoi SSR, 21, 66-72 (1972) · Zbl 0227.62019
[25] Kuks, J., Minimax estimation of regression coefficients (Russian), Izvestija Akademii Nauk Estonskoi SSR, 21, 73-78 (1972) · Zbl 0227.62020
[26] La Motte, L. R., Bayes linear estimators, Technometrics, 20, 281-290 (1978) · Zbl 0399.62070
[27] Läuter, H., A minimax linear estimator for linear parameters under restrictions in form of inequalities, Math. Operationsforsch. Statist. Ser. Statist., 6, 689-696 (1975) · Zbl 0331.62048
[28] O’Hagan, A., Bayes estimation of a convex quadratic, Biometrika, 60, 565-571 (1973) · Zbl 0277.62052
[29] Oman, S. D., Regression estimation for a bounded response over a bounded region, Technometrics, 25, 251-261 (1983) · Zbl 0526.62063
[30] Pilz, J., Bayesian Estimation and Experimental Design in Linear Regression Models, (Teubner-Texte, Vol. 55 (1983), Teubner: Teubner Leipzig) · Zbl 0533.62007
[31] Pilz, J., Robust Bayes regression estimation under weak prior knowledge, (Rasch, D.; Tiku, M. L., Robustness of Statistical Methods and Nonparametric Statistics (1984), Deutscher Verlag der Wissenschaften: Deutscher Verlag der Wissenschaften Berlin) · Zbl 0595.62065
[32] Pilz, J., A note on Krafft’s maximin linear estimator for linear regression parameters, Statistics (1984), to appear · Zbl 0587.62139
[33] Pilz, J., Minimax straight line regression estimation using prior bounds for the parameters, (Bandemer, H., Freiberger Forschungshefte D170 (1985), Deutscher Verlag für Grundstoffindustrie: Deutscher Verlag für Grundstoffindustrie Leipzig) · Zbl 0593.62061
[34] Pukelsheim, F., On linear regression designs which maximize information, J. Statist. Plann. Inference, 4, 339-364 (1980) · Zbl 0472.62079
[35] Rao, C. R., Linear Statistical Inference and its Applications (1965), Wiley: Wiley New York · Zbl 0137.36203
[36] Rao, C. R., Estimation of parameters in a linear model, Ann. Statist., 4, 1023-1037 (1976) · Zbl 0336.62055
[37] Schmidt, P., Constraints on the parameters in simultaneous tobit and probit models, (Manski, C. F.; McFadden, D., Structural Analysis of Discrete Data, with Econometric Applications (1981), MIT Press: MIT Press Cambridge) · Zbl 0527.62099
[38] Silvey, S. D., Optimal Design (1980), Chapman and Hall: Chapman and Hall London-New York · Zbl 0468.62070
[39] Sion, M., On general minimax theorems, Pacific J. Math., 8, 171-176 (1958) · Zbl 0081.11502
[40] Teräsvirta, T., Restricted superiority of linear homogeneous estimators over ordinary least squares, Scand. J. Statist., 10, 27-33 (1983) · Zbl 0523.62065
[41] Theil, H.; Goldberger, A. S., On pure and mixed estimation in economics, Internat. Econom. Rev., 2, 65-78 (1961)
[42] Toutenburg, H., Prior Information in Linear Models (1982), Wiley: Wiley Chichester · Zbl 0468.62056
[43] Toutenburg, H.; Roeder, B., Minimax-linear and Theil estimator for restrained regression coefficients, Math. Operationsforsch. Statist. Ser. Statist., 9, 499-505 (1978) · Zbl 0426.62043
[44] Trenkler, G.; Trenkler, D., A note on superiority comparisons of homogeneous linear estimators, Comm. Statist. A, 12, 799-808 (1983) · Zbl 0523.62066
[45] Whittle, P., Some general points in the theory of optimal experimental design, J. Roy. Statist. Soc. B, 35, 123-130 (1973) · Zbl 0282.62065
[46] Wu, C. F., Some algorithmic aspects of the theory of optimal designs, Ann. Statist., 6, 1286-1301 (1978) · Zbl 0392.62058
[47] Wu, C. F.; Wynn, H. P., The convergence of general step-length algorithms for regular optimum design criteria, Ann. Statist., 6, 1273-1285 (1978) · Zbl 0396.62059
[48] Zacks, S., The Theory of Statistical Inference (1971), Wiley: Wiley New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.