The Annals of Statistics

Robust designs for polynomial regression by maximizing a minimum of D- and D1-efficiencies

Holger Dette and Tobias Franke

Full-text: Open access

Abstract

In the common polynomial regression of degree m we determine the design which maximizes the minimum of the $D$-efficiency in the model of degree $m$ and the $D_1$-efficiencies in the models of degree $m-j,\dots, m +k$ ($j, k\ge 0$ given). The resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness-of-fit of the assumed model of degree $m$ by testing the highest coefficients in the polynomials of degree $m-j,\dots, m +k$ .

Our approach is based on a combination of the theory of canonical moments and general equivalence theory for minimax optimality criteria. The optimal designs can be explicitly characterized by evaluating certain associated orthogonal polynomials.

Article information

Source
Ann. Statist., Volume 29, Number 4 (2001), 1024-1049.

Dates
First available in Project Euclid: 14 February 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1013699990

Digital Object Identifier
doi:10.1214/aos/1013699990

Mathematical Reviews number (MathSciNet)
MR1869237

Zentralblatt MATH identifier
1012.62080

Subjects
Primary: 62K05: Optimal designs 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Keywords
Minimax optimal designs robust design D-optimality D_1-optimality t-test, associated orthogonal polynomials

Citation

Dette, Holger; Franke, Tobias. Robust designs for polynomial regression by maximizing a minimum of D - and D 1 -efficiencies. Ann. Statist. 29 (2001), no. 4, 1024--1049. doi:10.1214/aos/1013699990. https://projecteuclid.org/euclid.aos/1013699990


Export citation

References

  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. Dover, New York.
  • 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.
  • Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
  • Dette, H. (1990). A generalization of Dand D1-optimal design in polynomial regression. Ann. Statist. 18 1784-1804.
  • Dette, H. (1995). Optimal designs for identifying the degree of a polynomial regression. Ann. Statist. 23 1248-1266.
  • Dette, H. and Studden, W. J. (1997). The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis. Wiley, New York.
  • Franke, T. (2000). Du nd D1-optimale Versuchspl¨ane unter Nebenbedingungen und gewichtete Maximin-Versuchspl¨ane bei polynomialer Regression. Dissertation, RuhrUniv. Bochum (in German).
  • Grosjean, C. C. (1986). The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials. J. Comp. Appl. Math. 18 259-307.
  • Guest, P. G. (1958). The spacing of observations in polynomial regression. Ann. Math. Statist. 29 294-299.
  • Hoel, P. G. (1958). Efficiency problems in polynomial estimation. Ann. Math. Statist. 29 1134-1145.
  • Huber, P. J. (1975). Robustness and designs. In A Survey of Statistical Designs (J. N. Srivastava, ed.) 287-301. North Holland, Amsterdam.
  • Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849-879.
  • Kiefer, J. C. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271-294.
  • Lasser, R. (1994). Orthogonal polynomials and hypergroups II: the symmetric case. Trans. Amer. Math. Soc. 341 749-770.
  • Lau, T. S. (1983). Theory of canonical moments and its applications in polynomial regression I, II. Technical Reports 83-23, 83-24, Purdue Univ.
  • Lau, T. S. (1988). D-optimal designs on the unit q-ball. J. Statist. Plann. Inference 19 299-315.
  • L¨auter, E. (1974). Experimental design in a class of models. Math. Oper. Statist. 5 379-398.
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
  • Pukelsheim, F. and Studden, W. J. (1993). E-optimal designs for polynomial regression. Ann. Statist. 21 402-415.
  • Silvey, S. D. (1980). Optimal Design. Chapman and Hall, London.
  • Skibinsky, M. (1967). The range of the n + 1 th moment for distributions on 0 1. J. Appl. Probab. 4 543-552.
  • Skibinsky, M. (1986). Principal representations and canonical moment sequences for distributions on an interval. J. Math. Anal. Appl. 120 95-120.
  • Spruill, M. G. (1990). Good designs for testing the degree of a polynomial mean. Sankhy¯a Ser. B 52 67-74.
  • Stigler, S. (1971). Optimal experimental design for polynomial regression. J. Amer. Statist. Assoc. 66 311-318.
  • Studden, W. J. (1968). Optimal designs on Tchebycheff points. Ann. Math. Statist. 39 1435-1447.
  • Studden, W. J. (1980). Ds-optimal designs for polynomial regression using continued fractions. Ann. Statist. 8 1132-1141. Studden, W. J. (1982a). Optimal designs for weighted polynomial regression using canonical moments. In Statistical Decision Theory and Related Topics III 2 (S. S. Gupta and J. O. Berger, eds.) 335-350. Academic Press, New York. Studden, W. J. (1982b). Some robust-type D-optimal designs in polynomial regression. J. Amer. Statist. Assoc. 77 916-921.
  • Studden, W. J. (1989). Note on some p-optimal designs for polynomial regression. Ann. Statist. 17 618-623.
  • Szeg ¨o, G. (1975). Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ., Providence, RI.
  • Wiens, D. P. (1992). Minimax designs for approximately linear regression. J. Statist. Plann. Inference 31 353-371.
  • Wong, W. K. (1994). Comparing robust properties of A-, D-, Eand G-optimal designs. Comput. Statist. Data Anal. 18 441-448.