Sampling designs for estimation of a random process

A random process X(t), t[epsilon][0,1], is sampled at a finite number of appropriately designed points. On the basis of these observations, we estimate the values of the process at the unsampled points and we measure the performance by an integrated mean square error. We consider the case where the process has a known, or partially or entirely unknown mean, i.e., when it can be modeled as X(t) = m(t) + N(t), where m(t) is nonrandom and N(t) is random with zero mean and known covariance function. Specifically, we consider (1) the case where m(t) is known, (2) the semiparametric case where m(t) = [beta]1[latin small letter f with hook]1(t)+...+[beta]q[latin small letter f with hook]fq(t), the [beta]i's are unknown coefficients and the [latin small letter f with hook]i's are known regression functions, and (3) the nonparametric case where m(t) is unknown. Here fi(t) and m(t) are of comparable smoothness with the purely random part N(t), and N(t) has no quadratic mean derivative. Asymptotically optimal sampling designs are found for cases (1), (2) and (3) when the best linear unbiased estimator (BLUE) of X(t) is used (a nearly BLUE in case (3)), as well as when the simple nonparametric linear interpolator of X(t) is used. Also it is shown that the mean has no effect asymptotically, and several examples are considered both analytically and numerically.