Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t [epsilon] T} be observable for each x in some compact set X. Assume that Y(x, t) = [theta]0f0(x)(t) + ... + [theta]kfk(x)(t) + N(t) where fi are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with the closed convex hull of the set {([phi], f(x))H : ||[phi] ||H <= 1, x [epsilon] X}, where (·, ·)H is the inner product on H. It is shown that if X is convex and fi are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c'[theta] can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.