Remez's Procedure for Finding Optimal Designs

Abstract

The Remez exchange procedures of approximation theory are used to find the optimal design for the problem of estimating $c'\theta$ in the regression model $EY(x) = \theta_1f_1(x) + \theta_2f_2(x) + \cdots + \theta_kf_k(x)$, when $c$ is not a linear combination of less than $k$ vectors of the form $f(x)$. A geometric approach is given first with a proof of convergence. When the design space is a closed interval, the Remez exchange procedure is illustrated by two examples. This type procedure can be used to find the optimal design very efficiently, if there exists an optimal design with $k$ support points.