Statistica Sinica 17(2007), 1005-1021

AN ALGEBRAIC CONSTRUCTION OF

MINIMALLY-SUPPORTED $\mbox{\boldmath $D$}$-OPTIMAL DESIGNS FOR

WEIGHTED POLYNOMIAL REGRESSION

Fu-Chuen Chang and Bo-Jung Jiang

National Sun Yat-sen University and

MassMutual Mercuries Life Insurance Co. Ltd.

Abstract: In this paper we investigate $(d+1)$-point $D$-optimal designs for $d$th degree polynomial regression with weight function $\omega(x)\ge 0$ on the interval $[a,b]$. We propose an algebraic approach and provide a numerical method for the construction of optimal designs. Thus if $\omega'(x)/\omega(x)$ is a rational function and the information of whether the optimal support contains the boundary points $a$ and $b$ is available, the problem of constructing $(d+1)$-point $D$-optimal designs can be transformed into a differential equation problem. One is led to a matrix that includes a finite number of auxiliary unknown constants, and the differentiation can be solved from a system of polynomial equations in those constants. Moreover, the $(d+1)$-point $D$-optimal interior support points are the zeros of a polynomial whose coefficients can be computed from a linear system.

Key words and phrases: Approximate D-optimal design, differential equation, matrix, minimally-supported, rational function, weighted polynomial regression.