A Weighting Scheme for Autoregressive Time Averages

Abstract

A simple arithmetic mean, when the observations are correlated, is not the optimum estimate of the population mean. Such is usually the case when the observations are taken from a stationary time series. The variances of the optimally weighted and the unweighted means, for different sample sizes, were compared when the data follow a second-order autoregressive scheme, and it was observed that, for small samples, much is to be gained if the observations are optimally weighted. In this paper, optimal weights are derived for the general case when the data follow an autoregressive scheme of order k, where k is any positive integer. These optimal weights are simple closed expressions in terms of the autoregressive coefficients.