Abstract
For the polynomial regression model on the interval [a, b] the optimal design problem with respect to Elfving's minimax criterion is considered. It is shown that the minimax problem is related to the problem of determining optimal designs for the estimation of the individual parameters. Sufficient conditions are given guaranteeing that an optimal design for an individual parameter in the polynomial regression is also minimax optimal for a subset of the parameters. The results are applied to polynomial regression on symmetric intervals [−b, b] (b≤1) and on nonnegative or nonpositive intervals where the conditions reduce to very simple inequalities, involving the degree of the underlying regression and the index of the maximum of the absolute coefficients of the Chebyshev polynomial of the first kind on the given interval. In the most cases the minimax optimal design can be found explicitly.
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References
Abramowitz, M. and Stegun, I. A. (1964).Handbook of Mathematical Functions, Dover Publications, New York.
Cantor, D. G. (1977). Solution to problem 6084 [1976,292],Amer. Math. Monthly,84, 832–833.
Davis, P. J. (1963).Interpolation and Approximation, Blaisdell Publishing, New York.
Dette, H. and Studden, W. J. (1992). Minimax designs in linear regression models, Tech. Report, #92-31, Dept. of Statistics, Purdue University, Indiana.
Elfving, G. (1952). Optimum allocation in linear regression theory,Ann. Math, Statist.,23, 255–262.
Elfving, G. (1959). Design of linear experiments,Probability and Statistics, The Harald Cramér Volume (ed. U. Grenander), 58–74, Almquist and Wiksell, Stockholm.
Fedorov, V. V. (1972).Theory of Optimal Experiments, Academic Press, New York.
Heiligers, B. (1992).E-Optimal Designs in Weighted Polynomial Regression, University of Aachen, Germany (preprint).
Kiefer, J. (1959). Optimum experimental designs,J. Roy. Statist. Soc. Ser. B,21, 272–304.
Murty, V. N. (1971). Minimax designs,J. Amer. Statist. Assoc.,66, 319–320.
Pazman, A. (1986).Foundations of Optimum Experimental Design, Reidel, Dordrecht.
Pukelsheim, F. (1993).Optimal Designs of Experiments, Wiley, New York.
Pukelsheim, F. and Studden, W. J. (1993).E-optimal designs for polynomial regression,Ann. Statist.,21, 401–415.
Rivlin, T. J. (1990).Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory, 2nd ed., Wiley-Interscience, New York.
Silvey, S. D. (1980).Optimal Design, Chapman and Hall, London.
Studden, W. J. (1968). Optimal designs on Tchebycheff points,Ann. Math. Statist.,39, 1435–1447.
Studden, W. J. (1982). Optimal designs for weighted polynomial regression using canonical moments,Statistical Decision Theory and Related Topics III, 335–350, Academic Press, New York.
Studden, W. J. and Tsay, J. Y. (1976). Remez's procedure for finding optimal designs,Ann. Statist.,4, 1271–1279.
Szegö, G. (1976). Orthogonal polynomials,Amer. Math. Soc. Colloq. Publ.,23, 4th ed.
Wong, W. K. (1992). A unified approach to the construction of minimax designs,Biometrika,79, 611–619.
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Research supported in part by the Deutsche Forschungsgemeinschaft.
Research supported in part by NSF Grant DMS 9101730.
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Dette, H., Studden, W.J. Optimal designs with respect to Elfving's partial minimax criterion in polynomial regression. Ann Inst Stat Math 46, 389–403 (1994). https://doi.org/10.1007/BF01720594
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DOI: https://doi.org/10.1007/BF01720594