Abstract
The extrapolation design problem for polynomial regression model on the design space [−1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs |ξ m * (z), ξ2m * (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atkinson, A. C. and Cox, D. R. (1974). Planning experiments for discriminating between models (with discussion), J. Roy. Statist. Soc. Ser. B, 36, 321–348.
Box, G. E. P. and Draper, N. R. (1959). A basis for the selection of a response surface design, J. Amer. Statist. Assoc., 54, 622–654.
Chao, M. T. (1995). Some robust type of extrapolation designs, Proceedings, International Conference on Statistical Methods and Statistical Computing and Productivity Improvement, 1, 99–108.
Dette, H. (1991). A note on robust designs for polynomial regression, J. Statist. Plann. Inference, 28, 223–232.
Dette, H. (1994). Discrimination designs for polynomial regression on a compact interval, Ann. Statist., 22, 890–904.
Dette, H. (1995a). Optimal designs for identifying the degree of a polynomial regression, Ann. Statist., 23, 1248–1267.
Dette, H. (1995b). A note on some peculiar extremal phenomena of the Chebyshev polynomials, Proc. Edinburgh Math. Soc., 38, 343–355.
Dette, H. and Studden, W. J. (1995). Optimal designs for polynomial regression when the degree of the polynomial is not known, Statist. Sinica, 5, 459–474.
Dette, H. and Studden, W. J. (1997). The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis, New York, Wiley.
Dette, H. and Wong, W. K. (1996). Robust optimal extrapolation designs, Biometrika, 83, 667–680.
Fedorov, V. V. (1972). Theory of Optimal Experiments, Academic Press, New York.
Hoel, P. G. (1965). Minimax designs in two dimensional regression, Ann. Math. Statist., 36, 1097–1106.
Hoel, P. G. and Levine, A. (1964). Optimal spacing and weighting in polynomial prediction, Ann. Math. Statist., 35, 1553–1560.
Huang, M.-N. L. and Studden, W. J. (1988). Model robust extrapolation designs, J. Statist. Plann. Inference, 13, 1–24.
Huber, P. J. (1975). Robustness and designs, A Survey of Statistical Design and Linear Models, 287–303, North Holland, Amsterdam.
Karlin, S. and Studden, W. J. (1966). Optimal experimental designs, Ann. Math. Statist., 37, 783–815.
Kiefer, J. C. (1980). Designs for extrapolation when bias is present, Multivariate Analysis-V: Proceedings of the Fifth International Symposium on Multivariate Analysis, North-Holland, Amsterdam.
Kiefer, J. and Studden, W. J. (1976). Optimal designs for large degree polynomial regression, Ann. Statist., 4, 1113–1123.
Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems, Ann. Math. Statist., 30, 271–294.
Kiefer, J. and Wolfowitz, J. (1965). On a theorem of Hoel and Levine on extrapolation, Ann. Math. Statist., 36, 1627–1655.
Lau, T. S. (1983). Theory of canonical moments and its application in polynomial I and II, Technical Reports, 83-23, 83-24, Department of Statistics, Purdue University, Lafayette, Indiana.
Läuter, E. (1974a). Experimental design for a class of models, Mathematische Operationsforschung und Statistik, 5, 379–398.
Läuter, E. (1974b). Optimal multipurpose designs for regression models, Mathematische Operationsforschung und Statistik, 7, 51–68.
Preitschopf, F. and Pukelsheim, F. (1987). Optimal designs for quadratic regression, J. Statist. Plann. Inference, 16, 213–218.
Pukelsheim, F. (1993). Optimal Design of Experiments, Wiley, New York.
Pukelsheim, F. and Rosenberger, J. L. (1993). Experimental designs for model discrimination, J. Amer. Statist. Assoc., 88, 642–649.
Rao, C. R. (1973). Linear Statistical Inference and Its Applications, Wiley, New York.
Pukelsheim, F. and Rosenberger, J. L. (1993). Experimental designs for model discrimination, J. Amer. Statist. Assoc., 88, 642–649.
Rao, C. R. (1973). Linear Statistical Inference and Its Applications, Wiley, New York.
Rivlin, T. J. (1990). Chebyshev Polynomials, Wiley, New York.
Sacks, J. and Ylvisaker, D. (1984). Some model robust designs in regression, Ann. Statist., 12, 1324–1348.
Spruill, M. C. (1984). Optimal designs for minimax extrapolation, J. Multivariate Anal., 15, 52–62.
Stigler, S. M. (1971). Optimal experimental design for polynomial regression, J. Amer. Statist. Assoc., 66, 311–318.
Studden, W. J. (1968). Optimal designs of Tchebyscheff points, Ann. Math. Statist., 39, 1435–1447.
Studden, W. J. (1971). Optimal designs for multivariate polynomial extrapolation, Ann. Math. Statist., 42, 828–832.
Studden, W. J. (1982). Some robust-type D-optimal designs in polynomial regression, J. Amer. Statist. Assoc., 77, 916–921.
Van Assche, W. (1987). Asymptotics for Orthogonal Polynomials, Lecture Notes in Mathematics, 1265, Springer, Berlin.
Author information
Authors and Affiliations
About this article
Cite this article
Dette, H., Lo Huang, MN. Convex Optimal Designs for Compound Polynomial Extrapolation. Annals of the Institute of Statistical Mathematics 52, 557–573 (2000). https://doi.org/10.1023/A:1004185822838
Issue Date:
DOI: https://doi.org/10.1023/A:1004185822838