SAMPLING DESIGNS FOR ESTIMATION OF A RANDOM PROCESS

A random process X(t), t is-an-element-of [0, 1], is sampled at a fini te number of appropriately designed points. On the basis of these obse rvations, we estimate the values of the process at the unsampled point s and we measure the performance by an integrated mean square error. W e consider the case where the process has a known, or partially or ent irely 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) = beta1f1(t) + ... beta(q)f(q)(t), the beta(i)'s are unknown coefficients and the f(i)'s are known regression functions, and (3) the nonparametric case where m(t) is unknown. Here f(i)(t) and m(t) are of comparable smoothness wi th the purely random part N(t), and N(t) has no quadratic mean derivat ive. 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 nonparame tric linear interpolator of X(t) is used. Also it is shown that the me an has no effect asymptotically, and several examples are considered b oth analytically and numerically.