Sampling from a stationary process and detecting a change in the mean of a stationary distribution

Abstract

Let $\left\{X_t\right\}$, $t\in R$ , be a stochastic process. Suppose that the process may not be continuously observed, yet an inference which is related to its probabilistic parameters, or to its sample path, is required. The main purpose of this paper is to study sampling plans. A sampling plan is a method for deciding about time instants $T_1,T_2,dots$ at which the process is observed. We study the effect of various sampling plans and sampling rates on the expected time to an alarm in change-point problems (of the mean). Our main effort is studying the asymptotic variance of the sum of the sampled observations until time t. This variance determines asymptotically the expected time to an alarm. As a by-product, we obtain the asymptotic variances of natural estimators for $p=E\left(X_t\right)$ and for $S_t={\int }_0^t{X}_{s}ds$ . Obviously, as the sampling rate is increased, a better estimation is possible. Our study enables us to decide on the `right' sampling rate. This is analogous to the problem of deciding on the `right' sample size in the case of independently and identically distributed observations.