Numerical methods for linear minimax estimation

• Autorzy: Norbert Gaffke , Berthold Heiligers
• Języki publikacji: EN

Abstrakty

EN

We discuss two numerical approaches to linear minimax estimation in linear models under ellipsoidal parameter restrictions. The first attacks the problem directly, by minimizing the maximum risk among the estimators. The second method is based on the duality between minimax and Bayes estimation, and aims at finding a least favorable prior distribution.

Słowa kluczowe

EN

Bayes estimation; duality; linear model; L-optimality; mean squared error; minimax estimation; non-smooth optimization; parameter restrictions; p-mean; quasi Newton method

Kategorie tematyczne

• 62J05: Linear regression
• 62F10: Point estimation

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae Probability and Statistics
• Rocznik: 2000
• Tom: 20
• Numer: 1
• Strony: 51-62

Daty

wydano

2000

otrzymano

1998-11-25

Twórcy

autor

Norbert Gaffke

• Fakultät für Mathematik, Institut für Mathematische Stochastik, Universität Magdeburg, D-39106 Magdeburg, Germany

autor

Berthold Heiligers

• Fakultät für Mathematik, Institut für Mathematische Stochastik, Universität Magdeburg, D-39106 Magdeburg, Germany

Bibliografia

• [1] W. Achtziger, Nichtglatte Optimierung: Ein spezielles Problem bei der Durchführung von Versuchen, Diplomathesis, Universität Bayreuth, Germany 1989 (in German).
• [2] A. Dietz, Implementierung eines Algorithmus zur Berechnung linearer Minimaxschätzer, Diplomathesis, Universität Augsburg, Germany 1993 (in German).
• [3] R. Fletcher, Practical Methods of Optimization, (2nd ed.) Wiley, New York 1987.
• [4] N. Gaffke and B. Heiligers, Bayes, admissible and minimax linear estimators in linear models with restricted parameter space, Statistics 20 (1989), 487-508.
• [5] N. Gaffke and B. Heiligers, Second order methods for solving extremum problems from optimal linear regression design, Optimization 36 (1996), 41-57.
• [6] N. Gaffke and R. Mathar, Linear minimax estimation and related Bayes L-optimal design, Methods of Operations Research 60 (1990), 617-628.
• [7] N. Gaffke and R. Mathar, On a class of algorithms from experimental design theory, Optimization 24 (1992), 91-126.
• [8] B. Heiligers, Affin-lineare Minimaxschätzer im linearen statistischen Modell bei eingeschränktem Parameterbereich, Diplomathesis, RWTH Aachen, Germany 1985 (in German).
• [9] B. Heiligers, Linear Bayes and minimax estimators in linear models with partially restricted parameter space, J. Statist. Plann. Inference 36 (1993), 175-183.
• [10] J.E. Higgins and E. Pollack, Minimizing pseudoconvex functions on convex compact sets, J. Optim. Theory and Appl. 65 (1990), 1-27.
• [11] K. Hoffmann, Characterization of minimax linear estimators in linear regression, Math. Operationsforsch. u. Statist., Ser. Statist. 8 (1979), 425-438.
• [12] J. Kuks, Minimax estimation of regression coefficients, Izv. Akad. Nauk Eston SSR 21 (1972), 73-78 (in Russian).
• [13] J. Lauterbach, Zur Berechnung approximativer Minimax-Schätzer im linearen Regressionsmodell, Dissertationthesis, Universität Hannover, Germany 1989 (in German).
• [14] J. Lauterbach and P. Stahlecker, Approximate minimax estimation in linear regression: A simulation study, Comm. Statist., Simulation 17 (1988), 209-227.
• [15] J. Pilz, Minimax linear regression estimation with symmetric parameter restrictions, J. Statist. Plann. Inference 13 (1986), 297-318.
• [16] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, Cambridge University Press, Cambridge 1988.
• [17] P. Stahlecker and J. Lauterbach, Approximate linear minimax estimation in regression analysis with ellipsoidal constraints, Comm. Statist., Theory Meth. 18 (1989), 2755-2784.

Identyfikatory

DOI

10.7151/dmps.1003