Statistica Sinica 18(2008), 1047-1062

OPTIMAL DESIGNS FOR FREE KNOT LEAST

SQUARES SPLINES

Holger Dette$^1$, Viatcheslav B. Melas$^2$ and Andrey Pepelyshev$^2$

$^1$Ruhr-Universität Bochum and $^2$St. Petersburg State University

Abstract: In this paper $D$-optimal designs for free knot least squares spline estimation are investigated. In contrast to most of the literature on optimal design for spline regression models it is assumed that the knots of the spline are also estimated from the data, which yields to optimal design problems for nonlinear models. In some cases local $D$-optimal designs can be found explicitly. Moreover, it is shown that the points of minimally supported $D$-optimal designs are increasing and real analytic functions of the knots; these results are used for the numerical construction of local $D$-optimal designs by means of Taylor expansions. In order to obtain optimal designs which are less sensitive to a specification of the unknown knots, a maximin approach is proposed and standardized maximin $D$-optimal designs for least square splines with estimated knots are determined in the class of all minimally supported designs.

Key words and phrases: Free knot least squares splines, D-optimal designs, nonlinear models, local optimal designs, robust designs, saturated designs.