On the Regression Design Problem of Sacks and Ylvisaker

Abstract

We consider the experimental design problem of Sacks and Ylvisaker. We consider only the case of the (noise) stochastic process $X$ satisfying a stochastic differential equation of the form \begin{equation*}\tag{0.1} L_mX(t) = dW(t)/dt\quad 0 \leqq t \leqq 1\end{equation*} where $L_m$ is an $m$th order differential operator whose null space is spanned by an ECT system and $W(t)$ is a Wiener process. We show that the non-degeneracy of the covariance matrix of $\{X^{(\nu)} (t_i), \nu = 0, 1, 2, \cdots, m - 1, t_i \in \lbrack 0, 1 \rbrack, i = 1,2, \cdots, n\}$ is equivalent to the total positivity properties of the Green's function for $L_m^\ast L_m$ with appropriate boundary conditions. An asymptotically optimal sequence of designs is found for this case and its dependence on the characteristic discontinuity of the above mentioned Green's function is exhibited. Finally we show that a special case of the problem is equivalent to the problem of the optimal approximation of a monomial by a Spline function in the $L_2$ norm. Some recent results are available on this latter problem which provide some information concerning existence and uniqueness of optimal designs with distinct points.