CONVEX OPTIMAL DESIGNS FOR COMPOUND POLYNOMIAL EXTRAPOLATION

HOLGER DETTE^{1} AND MONG-NA LO HUANG^{2}

1Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany 2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, R.O.C.

(Received May 18, 1998; revised May 13, 1999)

Abstract. The extrapolation design problem for polynomial regression model on the design space [−1, 1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees {m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs {ξ_{m}^{∗}(z), ξ_{2m}^{∗}(z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degrees {m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.

Key words and phrases: Chebyshev polynomials, convex combination, extremal problems for polynomials, Lagrange interpolation polynomial, optimal discrimination designs.