The optimal sampling design for estimating the integral of a process with stationary independent increments (Corresp.)

Abstract

Let I be the Riemann integral over [0,A] of a realization of a random process X(t) with stationary independent increments (SII), finite second moments, and X(0) = 0 . The optimal design, consisting of n distinct observation times, for estimating I relative to a mean-squared error optimality criterion is shown to be the design such that t_{i} = 2iA/(2n + 1), i = 1, \cdots,n , independent of the parameters of the process. The mean-squared error of the optimal estimate is displayed. The optimal design for estimating the integral over a proper subinterval of [0,A] is also derived.