On the estimation of the time mean-value of a stationary process

Abstract

This paper deals with the estimation of the time average 1 = nx (t) dt of a stationary stochastic process, {x (t); -00 < t < oo}, by means of a finite linear combination 1*of observed values from the interval [0,1]. A brief account of previous work on this problem is given, some general comments are made, and then the main part of the paper is devoted to the construction of an estimate 1*, which, asymptotically, minimizes E (1 -1*)2, to'" t n being equidistant (except in the case of a firstorder Markov process). This 'best' estimate turns out to be, to a certain extent, independent of the spectral properties of the process.