, Volume 46, Issue 1, pp 71–82 | Cite as

A note on the uniform distribution on the arcsin points

  • Holger Dette


In his book Pukelsheim [8] pointed out that designs supported at the arcsin points are very efficient for the statistical inference in a polynomial regression model. In this note we determine the canonical moments of a class of distributions which have nearly equal weights at the arcsin points. The class contains theD-optimal arcsin support design and theD 1-optimal design for a polynomial regression. The results allow explicit representations ofD-, andD 1-efficiencies of these designs in all polynomial models with a degree less than the number of support points of the design.

Keywords and Phrases

arcsin support designs canonical moments D-efficiency 


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  1. [1]
    Anderson TW (1962) The choice of the degree in a polynomial regression as a multiple decision problem. Ann. Math. Statist. 33:255–265MathSciNetGoogle Scholar
  2. [2]
    Chihara TS (1978) Introduction to orthogonal polynomials. Gordon and Breach, New YorkMATHGoogle Scholar
  3. [3]
    Dette H (1994) Discrimination designs for polynomial regression on compact intervals. Ann. Statist. 22:890–903MATHMathSciNetGoogle Scholar
  4. [4]
    Dette H, Studden WJ (1997) The theory of canonical moments with applications in statistics, probability and analysis. Wiley, New YorkMATHGoogle Scholar
  5. [5]
    Karlin, Studden WJ (1966) Tchebycheff systems: With applications in analysis and statistics. Interscience Publishers, Wiley, New YorkMATHGoogle Scholar
  6. [6]
    Kiefer J (1974) General equivalance theory for optimum designs (approximate theory). Ann. Statist. 2:849–879MATHMathSciNetGoogle Scholar
  7. [7]
    Läuter E (1974) Experimental design in a class of models. Math. Operations-forschung Statist., Ser. Statist. 5:379–398Google Scholar
  8. [8]
    Pukelsheim F (1993) Optimal design of experiments. Wiley, New YorkMATHGoogle Scholar
  9. [9]
    Pukelsheim F, Studden WJ (1993) E-optimal designs for polynomial regression. Ann. Statist. 21:402–415MATHMathSciNetGoogle Scholar
  10. [10]
    Skibinsky M (1986) Principal representations and canonical moment sequences for distributions on an interval. J. Math. Appl. 120:95–120MATHMathSciNetGoogle Scholar
  11. [11]
    Studden WJ (1968) Optimal designs on Tchebycheff points. Ann. Math. Statist. 39:1435–1447MathSciNetGoogle Scholar
  12. [12]
    Studden WJ (1980)D s-optimal designs for polynomial regression using continued fractions. Ann. Statist. 8:1132–1141MATHMathSciNetGoogle Scholar
  13. [13]
    Studden WJ (1982) Optimal designs for weighted polynomial regression using canonical moments. In: Statistical decision theory and related topics III, Gupta SS, Berger JO (eds.) Academic Press, New York:335–350Google Scholar

Copyright information

© Physica-Verlag 1997

Authors and Affiliations

  • Holger Dette
    • 1
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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