Abstract
In the common trigonometric regression model we investigate the D-optimal design problem, where the design space is a partial circle. It is demonstrated that the structure of the optimal design depends only on the length of the design space and that the support points (and weights) are analytic functions of this parameter. By means of a Taylor expansion we provide a recursive algorithm such that the D-optimal designs for Fourier regression models on a partial circle can be determined in all cases. In the linear and quadratic case the D-optimal design can be determined explicitly.
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References
Dette, H. (1998). Some applications of canonical moments in Fourier regression models, New Developments and Applications in Experimental Design (eds. N. Flournoy, W. F. Rosenberger and W. K. Wong), IMS Lecture Notes Monogr. Ser., 34, 175–185, Hayward, California.
Dette, H. and Haller, G. (1998). Optimal designs for the identification of the order of a Fourier regression, Ann. Statist., 26, 1496–1521.
Dette, H. and Melas, V. B. (2003). Optimal designs for estimating individual coefficients in Fourier regression models, Ann. Statist. (to appear), http://www.ruhr-uni-bochum.de/mathematik3/preprint.htm.
Dette, H. Melas, V. B. and Pepelyshev, A. (2002). Optimal designs for estimating individual coefficients—A functional approach, J. Statist. Plann. Inference (to appear), http://www.ruhr-uni-bochum.de/mathematik3/preprint.htm.
Fedorov, V. V. (1972). Theory of Optimal Experiments, Academic Press, New York.
Graybill, F. A. (1976). Theory and Application of the Linear Model, Wadsworth, Belmont, California.
Gunning, R. C. and Rossi, H. (1965). Analytical Functions of Several Complex Variables, Prentice Hall, New York.
Hill, P. D. H. (1978). A note on the equivalence of D-optimal design measures for three rival linear models, Biometrika, 65, 666–667.
Hoel, P. (1965). Minimax design in two-dimensional regression, Ann. Math. Statist., 36, 1097–1106.
Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, New York.
Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory), Ann. Statist., 2, 849–879.
Kitsos, C. P., Titterington, D. M. and Torsney, B. (1988). An optimal design problem in rhythmometry, Biometrics, 44, 657–671.
Lau, T. S. and Studden, W. J. (1985). Optimal designs for trigonometric and polynomial regression, Ann. Statist., 13, 383–394.
Mardia, K. (1972). The statistics of directional data, Academic Press, New York.
Melas, V. B. (1978). Optimal designs for exponential regression, Mathematische Operations for schung und Statistik, Series Statistics, 9, 45–59.
Pukelsheim, F. (1993). Optimal Design of Experiments, Wiley, New York.
Riccomagno, E., Schwabe, R. and Wynn, H. P. (1997). Lattice-based D-optimum design for Fourier regression, Ann. Statist., 25, 2313–2327.
Rivlin, T. J. (1974). Chebyshev Polynomials, Wiley, New York.
Silvey, S. D. (1980). Optimal Design, Chapman and Hall, London.
Szegö, G. (1975). Orthogonal polynomials, Amer. Math. Soc. Colloqu. Publ., 23, Providence, Rhode Island.
Wu, H. (2002). Optimal designs for first order trigonometric regression on a partial circle, Statistica Sinica, 12, 917–930.
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Dette, H., Melas, V.B. & Pepelyshev, A. D-Optimal Designs for Trigonometric Regression Models on a Partial Circle. Annals of the Institute of Statistical Mathematics 54, 945–959 (2002). https://doi.org/10.1023/A:1022436007242
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DOI: https://doi.org/10.1023/A:1022436007242