Optimal Designs for Approximating a Stochastic Process with Respect to a Minimax Criterion

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Abstract
We study the problem of approximating a stochastic process Y = {Y(t: t ∈ T} with known and continuous covariance function R on the basis of finitely many observations Y(t 1,), …, Y(t n ). Dependent on the knowledge about the mean function, we use different approximations Ŷ and measure their performance by the corresponding maximum mean squared error sub t∈T E(Y(t) − Ŷ(t))2. For a compact T ⊂ ℝ p we prove sufficient conditions for the existence of optimal designs. For the class of covariance functions on T 2 = [0, 1]2 which satisfy generalized Sacks/Ylvisaker regularity conditions of order zero or are of product type, we construct sequences of designs for which the proposed approximations perform asymptotically optimal.

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