Abstract
This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.
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References
Biedermann S, Dette H, Hoffmann P (2009) Constrained optimal discrimination designs for Fourier regression models. Ann Inst Stat Math 61(1):143–157
Box G, Draper N (1959) A basis for the selection of a response surface design. J Am Stat Assoc 54:622–654
Cartwright M (1990) Fourier methods for mathematicians, scientists, and engineers. Ellis Horwood, New York
Dette H, Haller G (1998) Optimal designs for the identification of the order of a Fourier regression. Ann Stat 26:1496–1521
Dette H, Melas VB (2003) Optimal designs for estimating individual coefficients in Fourier regression models. Ann Stat 31:1669–1692
Dette H, Melas VB, Pepelyshev A (2002) D-optimal designs for trigonometric regression models on a partial circle. Ann Inst Stat Math 54(4):945–959
Dette H, Melas VB, Pepelyshev A (2004) Optimal designs for estimating individual coefficients—a functional approach. J Stat Plan Inference 118:201–219
Dette H, Melas VB, Shpilev P (2007) Optimal designs for estimating coefficients of the lower frequencies in trigonometric regression models. Inst Stat Math 59:655–673
Dette H, Melas VB, Shpilev P (2009) Optimal designs for estimating pairs of coefficients in Fourier regression models. Stat Sin 19:1587–1601
Dette H, Melas VB, Shpilev P (2011) Optimal designs for trigonometric regression models. J Stat Plan Inference 141:1343–1353
Hoel PG (1965) Minimax designs in two dimensional regression. Ann Math Stat 36:1097–1106
Heo G, Schmuland B, Wiens DP (2001) Restricted minimax robust designs for misspecified regression models. Can J Stat 29:117–128
Lestrel PE (1997) Fourier descriptors and their applications in biology. Cambridge University Press, Cambridge
Kiefer J (1974) General equivalence theory for optimum designs (approximate theory). Ann Stat 2:849–879
Kiefer J, Wolfowitz J (1960) The equivalence of two extremum problems. Can J Math 12:363–366
Kupper LL (1973) Minimax designs for Fourier series and Sphericak harmonics regressions: a characterization of rotatable arrangements. J R Stat Soc 35:493–500
Kitsos CP, Titterington DM, Torsney B (1988) An optimal design problem in rhythmometry. Biometrics 44:657–671
McCool JI (1979) Systematic and random errors in least squares estimation for circular contours. Precis Eng 1:215–220
Pukelsheim F (1993) Optimal design of experiments. Wiley, New York
Shi P, Ye JJ, Zhou J (2003) Minimax robust designs for misspecified regression models. Can J Stat 31: 397–414
Wiens DP, Zhou J (1996) Minimax regression designs for approximately linear models with autocorrelated errors. J Stat Plan Inference 55:95–106
Xu X (2009) Robust prediction and extrapolation designs for censored data. J Stat Plan Inference 139: 486–502
Xu X and Shang X (2013) D-optimal designs for full and reduced Fourier regression models. Manuscript. Available at http://spartan.ac.brocku.ca/xxu/Fourier_D_optimal.pdf
Xu X, Zhao L (2009) Robust designs for haar wavelet approximation models. Appl Stoch Models Bus Ind 27(5):531–550
Ye J, Zhou J (2007) Existence and symmetry of minimax regression designs. J Stat Plan Inference 137: 344–354
Acknowledgments
This research is supported by the Natural Sciences and Engineering Research Council of Canada. The authors are thankful to the editor and two anonymous referees for their insightful comments.
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Xu, X., Shang, X. Optimal and robust designs for trigonometric regression models. Metrika 77, 753–769 (2014). https://doi.org/10.1007/s00184-013-0463-7
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DOI: https://doi.org/10.1007/s00184-013-0463-7