On prediction of moving-average processes

Let {X<inf>n</inf>} be a discrete-time stationary moving-average process having the representation where the real-valued process (Y<inf>n</inf>) has a well-defined entropy and spectrum. Let ∊^∗2<inf>k</inf> denote the smallest mean-squared error of any estimate of X<inf>n</inf> based on observations of X<inf>n–1</inf>, X<inf>n–2</inf>, …, X<inf>n–k</inf>, and let ∊^∗2<inf>klin</inf>, be the corresponding least mean-squared error when the estimator is linear in the k observations. We establish an inequality of the form where G(Y) ≤ 1 depends only on the entropy and spectrum of {Y<inf>n</inf>}. We also obtain explicit formulas for ∊^∗2<inf>k</inf> and ∊^∗2<inf>klin</inf> and compare these quantities graphically when M = 2 and the {Y<inf>n</inf>} are i.i.d. variates with one of several different distributions. The best estimators are quite complicated but are frequently considerably better than the best linear ones. This extends a result of M. Kanter.