AISM 54, 945-959

© 2002 ISM

*D*-optimal designs for trigonometric regression models on a partial circle

### Holger Dette^{1}, Viatcheslav B. Melas^{2} and Andrey Pepelyshev^{2}

^{1}Fakultät für Mathematik, Lehrstuhl III
(Stochastik), Ruhr-Universität Bochum, Gebaude NA 3/72, 44780 Bochum, Germany, e-mail: holger.dette@ruhr-uni-bochum.de

^{2}Department of Mathematics, St. Petersburg State University,
St. Petersburg, Russian Federation, e-mail: v.melas@pobox.spbu.ru; andrey@ap7236.spb.edu

(Received March 16, 2001; revised August 24, 2001)

Abstract.
In the common trigonometric regression model we investigate the $D$-optimal design problem, where the design space is a partial circle. It is demonstrated that the structure of the optimal design depends only on the length of the design space and that the support points (and
weights) are analytic functions of this parameter. By means of a Taylor expansion
we provide a recursive algorithm such that the $D$-optimal designs for Fourier regression models
on a partial circle can be determined in all cases. In the linear and quadratic case the $D$-optimal design can be determined explicitly.

Key words and phrases:
Trigonometric regression, D-optimality, implicit function theorem, orthogonal polynomial.

**Source**
(TeX , DVI )