Some explicit solutions of c-optimal design problems for polynomial regression with no intercept

  • Holger DetteEmail author
  • Viatcheslav B. Melas
  • Petr Shpilev


In this paper, we consider the optimal design problem for extrapolation and estimation of the slope at a given point, say z, in a polynomial regression with no intercept. We provide explicit solutions of these problems in many cases and characterize those values of z, where this is not possible.


Polynomial regression Extrapolation Slope estimation c-optimal designs 



This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt C2) of the German Research Foundation (DFG). The work of Viatcheslav Melas and Petr Shpilev was partly supported by Russian Foundation for Basic Research (Project No. 20-01-00096). The authors thank Martina Stein, who typed parts of this manuscript with considerable technical expertise. The authors are also grateful to two referees and an associate editor for their constructive comments on an earlier version of this paper.


  1. Celant, G., Broniatowski, M. (2016). Interpolation and extrapolation designs for the polynomial regression, chapter 4, pp. 69–111. New York: Wiley.CrossRefGoogle Scholar
  2. Chang, F.-C., Heiligers, B. (1996). E-optimal designs for polynomial regression without intercept. Journal of Statistical Planning and Inference, 55(3), 371–387.MathSciNetCrossRefGoogle Scholar
  3. Chang, F.-C., Lin, G.-C. (1997). \(D\)-optimal designs for weighted polynomial regression. Journal of Statistical Planning and Inference, 62, 317–331.MathSciNetCrossRefGoogle Scholar
  4. Dette, H. (1990). A generalization of \(D\)- and \(D_1\)-optimal designs in polynomial regression. Annals of Statistics, 18, 1784–1805.MathSciNetCrossRefGoogle Scholar
  5. Dette, H., Franke, T. (2001). Robust designs for polynomial regression by maximizing a minimum of \(D-\) and \(D_1\)-efficiencies. Annals of Statistics, 29(4), 1024–1049.MathSciNetCrossRefGoogle Scholar
  6. Dette, H., Huang, M.-N. L. (2000). Convex optimal designs for compound polynomial extrapolation. Annals of the Institute of Statistical Mathematics, 52(3), 557–573.MathSciNetCrossRefGoogle Scholar
  7. Dette, H., Melas, V. B., Pepelyshev, A. (2004). Optimal designs for estimating individual coefficients in polynomial regressiona functional approach. Journal of Statistical Planning and Inference, 118(1), 201–219.MathSciNetCrossRefGoogle Scholar
  8. Dette, H., Melas, V. B., Pepelyshev, A. (2010). Optimal designs for estimating the slope of a regression. Statistics, 44(6), 617–628.MathSciNetCrossRefGoogle Scholar
  9. Dette, H., Studden, W. J. (1993). Geometry of \(E\)-optimality. Annals of Statistics, 21(1), 416–433.MathSciNetCrossRefGoogle Scholar
  10. Dette, H., Wong, W. K. (1996). Robust optimal extrapolation designs. Biometrika, 83(3), 667–680.MathSciNetCrossRefGoogle Scholar
  11. Elfving, G. (1952). Optimal allocation in linear regression theory. The Annals of Mathematical Statistics, 23, 255–262.CrossRefGoogle Scholar
  12. Fang, Z. (2002). \(D\)-optimal designs for polynomial regression models through origin. Statistics and Probability Letters, 57, 343–351.MathSciNetCrossRefGoogle Scholar
  13. Hoel, P. G. (1958). Efficiency problems in polynomial estimation. Annals of Mathematical Statistics, 29(4), 1134–1145.MathSciNetCrossRefGoogle Scholar
  14. Hoel, P. G., Levine, A. (1964). Optimal spacing and weighting in polynomial prediction. Annals of Mathematical Statistics, 35(4), 1553–1560.MathSciNetCrossRefGoogle Scholar
  15. Huang, M.-N. L., Chang, F.-C., Wong, W. K. (1995). \(D\)-optimal designs for polynomial regression without an intercept. Statistica Sinica, 5(2), 441–458.MathSciNetzbMATHGoogle Scholar
  16. Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). The Annals of Statistics, 2(5), 849–879.MathSciNetCrossRefGoogle Scholar
  17. Mandal, N., Heiligers, B. (1992). Minimax designs for estimating the optimum point in a quadratic response surface. Journal of Statistical Planning and Inference, 31, 235–244.MathSciNetCrossRefGoogle Scholar
  18. Melas, V., Pepelyshev, A., Cheng, R. (2003). Designs for estimating an extremal point of quadratic regression models in a hyperball. Metrika, 58, 193–208.MathSciNetCrossRefGoogle Scholar
  19. Pronzato, L., Walter, E. (1993). Experimental design for estimating the optimum point in a response surface. Acta Applicandae Mathematicae, 33, 45–68.MathSciNetCrossRefGoogle Scholar
  20. Pukelsheim, F. (2006). Optimal design of experiments. Philadelphia: SIAM.CrossRefGoogle Scholar
  21. Pukelsheim, F., Studden, W. J. (1993). \(E\)-optimal designs for polynomial regression. Annals of Statistics, 21(1), 402–415.MathSciNetCrossRefGoogle Scholar
  22. Sahm, M. (1998). Optimal designs for estimating individual coefficients in polynomial regression. Ph.D. thesis, Fakultät für Mathematik, Ruhr-Universität Bochum, Germany.Google Scholar
  23. Studden, W. J. (1968). Optimal designs on Tchebycheff points. Annals of Mathematical Statistics, 39(5), 1435–1447.MathSciNetCrossRefGoogle Scholar
  24. Zen, M.-M., Tsai, M.-H. (2004). Criterion-robust optimal designs for model discrimination and parameter estimation in Fourier regression models. Journal of Statistical Planning and Inference, 124, 475–487.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  • Holger Dette
    • 1
    Email author
  • Viatcheslav B. Melas
    • 2
  • Petr Shpilev
    • 2
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Mathematics and Mechanics FacultySt. Petersburg State UniversitySt. PetersburgRussia

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