Statistica Sinica 9(1999), 1053-1069

ROBUST DESIGNS FOR FITTING LINEAR MODELS

WITH MISSPECIFICATION

Rong-Xian Yue$^{*\dagger}$ and Fred J. Hickernell$^*$

$^*$Hong Kong Baptist University and $^\dagger$Shanghai Normal University

Abstract: This paper considers linear models with misspecification of the form $f({\bf x})=E(y\vert{\bf x})=\sum_{j=1}^{p} \theta_j g_j({\bf x})+h({\bf x})$, where $h({\bf x})$ is an unknown function. We assume that the true response function $f$ comes from a reproducing kernel Hilbert space and the estimates of the parameters $\theta_j$ are obtained by the standard least squares method. A sharp upper bound for the mean squared error is found in terms of the norm of $h$. This upper bound is used to choose a design that is robust against the model bias. It is shown that the continuous uniform design on the experimental region is the all-bias design. The numerical results of several examples show that all-bias designs perform well when some model bias is present in low dimensional cases.

Key words and phrases: Linear models with misspecification, model-robust designs, reproducing kernel Hilbert spaces.