Abstract
In linear regression models, predictors based on least squares or on generalized least squares estimators are usually applied which, however, fail in case of multicollinearity. As an alternative biased estimators like ridge estimators, Kuks-Olman estimators, Bayes or minimax estimators are sometimes suggested. In our analysis the relative instead of the generally used absolute squared error enters the objective function. An explicit minimax solution is derived which, in an important special case, can be viewed as a predictor based on a Kuks-Olman estimator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, B.F., Stahlecker, P. (1998): Prediction in Linear Regression Combining Cris Data and Fuzzy Prior Information. Statistics & Decisions 16, 19–33.
Arnold, B.F., Stahlecker, P. (2000): Another View of the Kuks-Olman Estimator. Journa of Statistical Planning and Inference 89, 169–174.
Christopeit, N., Girko, V.L. (1995): Minimax Estimators for Linear Models with Non Random Disturbances. Random Operators and Stochastic Equations 3, 361–376.
Drygas, H. (1975): Estimation and Prediction for Linear Models in General Spaces Mathematische Operationsforschung und Statistik, Series Statistics 6, 301–324.
Gaffke, N., Heiligers, B. (1989): Bayes, Admissible, and Linear Minimax Estimators i: Linear Models with Restricted Parameter Space. Statistics 20, 487–508.
Goldberger, A.S. (1962): Best Linear Unbiased Prediction in the Generalized Linea Regression Model. Journal of the American Statistical Association 57, 369–375.
Hoerl, A. E., Kennard, R. W. (1970): Ridge Regression: Biased Estimation of Nonorthog onal Problems. Technometrics 12, 55–67.
Hoffmann, K. (1979): Characterization of Minimax Linear Estimators in Linear Regres sion. Mathematische Operationsforschung und Statistik, Series Statistics 10, 19–26.
Kuks, J., Olman, W. (1971): Minimax Linear Estimation of Regression Coefficients I (i: Russian). Iswestija Akademija Nauk Estonskoj SSR 20, 480–482.
Kuks, J., Olman, W. (1972): Minimax Linear Estimation of Regression Coefficients] (in Russian). Iswestija Akademija Nauk Estonskoj SSR 21, 66–72.
Läuter, H. (1975): A Minimax Linear Estimator for Linear Parameters Under Restric tions in Form of Inequalities. Mathematische Operationsforschung und Statistik, Serie Statistics 6, 689–695.
Lauterbach, J., Stahlecker, P. (1990): Some Properties of [tr(Q 2p)]1/2p with Application to Linear Minimax Estimation. Linear Algebra and its Applications 127, 301–325.
Pilz, J. (1986): Minimax Linear Regression Estimation with Symmetric Parameter Restriction. Journal of Statistical Planning and Inference 13, 297–318.
Pilz, J. (1991): Bayesian Estimation and Experimental Design in Linear Regression Models. New York (Wiley).
Pukelsheim, F. (1993): Optimal Design of Experiments. New York (Wiley).
Rao, C. R., Toutenburg, H. (1995): Linear Models, Least Squares and Alternatives. Heidelberg (Springer).
Stahlecker, P. (1987): A priori Information und Minimax-Schätzung im Linearen Regressionsmodell (in German). Mathematical Systems in Economics 108. Frankfurt (Athenäum).
Stahlecker, P., Trenkler, G. (1993): Minimax Estimation in Linear Regression with Singular Covariance Structure and Convex Polyhedral Constraints. Journal of Statistical Planning and Inference 36, 185–196.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arnold, B.F., Stahlecker, P. Relative squared error prediction in the generalized linear regression model. Statistical Papers 44, 107–115 (2003). https://doi.org/10.1007/s00362-002-0136-5
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s00362-002-0136-5