Optimal designs for estimating the path of a stochastic process

Bhramar Mukherjee, Purdue University

Abstract

A second-order random process Y(t), with E(Y(t)) ≡ 0, is sampled at a finite number of design points t1, t2,…,tn. On the basis of these observations, one wants to estimate the values of the process at unsampled points using the best linear unbiased estimator (BLUE). The performance of the estimator is measured by a weighted integrated mean squared error. The goal is to find t1,t 2,…,tn, such that this integrated mean squared error (IMSE) is minimized for a fixed n. This optimization problem depends on the stochastic process only through its covariance structure. For processes with a product type covariance structure, i.e., for Cov(Y(s), Y(t)) = u(s) v(t), s < t, we obtain a set of necessary and sufficient conditions for a design to be exactly optimal. Explicit calculations of optimal designs for any given n for Brownian Motion, Brownian Bridge and Ornstein-Uhlenbeck process illustrate the simplicity and usefulness of these conditions. Starting from the set of exact optimality conditions for a fixed n, an asymptotic result yielding the density whose percentile points furnish a set of asymptotically optimal design points (in some suitable sense) is derived. The problem of estimating the integral of Y(t) instead of the path is also considered. The integral estimation problem is related to certain regression design problems with correlated errors. The case when one tries to minimize the maximum mean squared error (MMSE) instead of the IMSE is discussed briefly. For a more general covariance structure, satisfying natural regularity conditions, some interesting asymptotic results are found. For processes with no quadratic mean derivative, a much simpler estimator is shown to be asymptotically equivalent to the BLUE. This leads to an intuitively appealing argument in establishing the asymptotic behavior of the BLUE and also in deriving an analytical expression for the asymptotically optimal design density.

Degree

Ph.D.

Advisors

Studden, Purdue University.

Subject Area

Statistics

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