Statistica Sinica 15(2005), 153-163

D-OPTIMAL DESIGNS FOR WEIGHTED POLYNOMIAL

REGRESSION - A FUNCTIONAL-ALGEBRAIC APPROACH

Fu-Chuen Chang

National Sun Yat-sen University

Abstract: This paper is concerned with the problem of computing the approximate $D$-optimal design for polynomial regression with weight function $\omega(x)> 0$ on the design interval $I=[m_0-a,m_0+a]$. It is shown that if $\omega'(x)/\omega(x)$ is a rational function on $I$ and $a$ is close to zero, then the problem of constructing $D$-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the $D$-optimal interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.

Key words and phrases: Approximate D-optimal design, implicit function theorem, matrix, rational function, recursive algorithm, Taylor series, weighted polynomial regression.