The Annals of Statistics

Optimal designs for the identification of the order of a Fourier regression

Holger Dette and Gerd Haller

Full-text: Open access

Abstract

For the Fourier regression model, we determine optimal designs for identifying the order of periodicity. It is shown that the optimal design problem for trigonometric regression models is equivalent to the problem of optimal design for discriminating between certain homo-and heteroscedastic polynomial regression models. These optimization problems are then solved using the theory of canonical moments, and the optimal discriminating designs for the Fourier regression model can be found explicitly. In contrast to many other optimality criteria for the trigonometric regression model, the optimal discriminating designs are not uniformly distributed on equidistant points.

Article information

Source
Ann. Statist., Volume 26, Number 4 (1998), 1496-1521.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1024691251

Digital Object Identifier
doi:10.1214/aos/1024691251

Mathematical Reviews number (MathSciNet)
MR1647689

Zentralblatt MATH identifier
0930.62075

Subjects
Primary: 62K05 62J05: Linear regression

Keywords
Fourier regression model optimal design model discrimination heteroscedastic polynomial regression canonical moments

Citation

Dette, Holger; Haller, Gerd. Optimal designs for the identification of the order of a Fourier regression. Ann. Statist. 26 (1998), no. 4, 1496--1521. doi:10.1214/aos/1024691251. https://projecteuclid.org/euclid.aos/1024691251


Export citation

References

  • ABRAMOWITZ, M. and STEGUN, I. A. 1964. Handbook of Mathematical Functions. Dover, New York. Z.
  • ANDERSON, T. W. 1962. The choice of the degree of a poly nomial regression as a multiple decision problem. Ann. Math. Statist. 33 255 265. Z.
  • ANDERSON, T. W. 1994. The Statistical Analy sis of Time Series. Wiley, New York. Z.
  • ATKINSON, A. C. 1972. Planning experiments to detect inadequate regression models. Biometrika 59 275 293. Z.
  • ATKINSON, A. C. and COX, D. R. 1974. Planning experiments for discriminating between models Z. with discussion. J. Roy. Statist. Soc. Ser. B 36 321 348. Z.
  • ATKINSON, A. C. and DONEV, A. N. 1992. Optimum Experimental Designs. Oxford Univ. Press. Z.
  • DETTE, H. 1994. Discrimination designs for poly nomial regression on a compact interval. Ann. Statist. 22 890 904.
  • DETTE, H. 1995. Optimal designs for identifying the degree of a poly nomial regression. Ann. Statist. 23 1248 1267.Z.
  • DETTE, H. and STUDDEN, W. J. 1997. The Theory of Canonical Moments with Applications in Statistics, Probability and Analy sis. Wiley, New York. Z.
  • FEDOROV, V. V. 1972. Theory of Optimal Experiments. Academic Press, New York. Z.
  • HILL, P. D. H. 1978. A note on the equivalence of D-optimal design measures for three rival linear models. Biometrika 65 666 667. Z.
  • HOEL, P. G. 1965. Minimax designs in two-dimensional regression. Ann. Math. Statist. 36 1097 1106. Z.
  • KARLIN, S. and STUDDEN, W. J. 1966. Tcheby cheff Sy stems: With Applications in Analy sis and Statistics. Interscience, New York. Z.
  • KIEFER, J. C. 1974. General equivalence theory for optimum designs. Ann. Statist. 2 849 879. Z.
  • LAU, T. S. 1983. Theory of canonical moments and its applications in poly nomial regression I, II. Technical Reports 83-23, 83-24, Purdue Univ. Z.
  • LAU, T. S. 1988. D-optimal designs on the unit q-ball. J. Statist. Plann. Inference 19 299 315. Z.
  • LAU, T. S. and STUDDEN, W. J. 1985. Optimal designs for trigonometric and poly nomial regression. Ann. Statist. 13 383 394. Z.
  • MARDIA, K. 1972. The Statistics of Directional Data. Academic Press, New York. Z.
  • PUKELSHEIM, F. 1993. Optimal Design of Experiments. Wiley, New York. Z.
  • RICCOMAGNO, E., SCHWABE, R., and Wy NN, H. P. 1997. Lattice-based D-optimum design for Fourier regression. Ann. Statist. 25 2313 2327 Z.
  • RIVLIN, T. J. 1990. Cheby shev Poly nomials. Wiley, New York. Z. Z.
  • SKIBINSKY, M. 1967. The range of the n 1 th moment for distributions on 0, 1. J. Appl. Probab. 4 543 552. Z.
  • SKIBINSKY, M. 1969. Some striking properties of binomial and beta moments. Ann. Math. Stast. 40 1753 1764. Z.
  • SKIBINSKY, M. 1986. Principal representations and canonical moment sequences for distributions on an interval. J. Math. Anal. Appl. 120 95 120. Z.
  • SPRUILL, M. G. 1990. Good designs for testing the degree of a poly nomial mean. Sankhy a Ser. B 52 67 74. Z.
  • STUDDEN, W. J. 1968. Optimal designs on Chebshev points. Ann. Math. Statist. 39 1435 1447. Z.
  • STUDDEN, W. J. 1980. D -optimal designs for poly nomial regression using continued fractions. s Ann. Statist. 8 1132 1141. Z.
  • STUDDEN, W. J. 1982a. Some robust-ty pe D-optimal designs in poly nomial regression. J. Amer. Statist. Assoc. 77 916 921. Z.
  • STUDDEN, W. J. 1982b. Optimal designs for weighted poly nomial regression using canonical Z moments. In Statistical Decision Theory and Related Topics 3 S. S. Gupta and J. O.. Berger, eds. 335 350. Academic, New York. Z.
  • STUDDEN, W. J. 1989. Note on some -optimal design for poly nomial regression. Ann. Statist. p 17 618 623. Z.
  • VAN ASSCHE, W. 1987. Asy mptotics for Orthogonal Poly nomials. Lecture Notes in Math. 1265. Springer, New York.