On prediction of moving-average processes

Let {Xn} be a discrete-time stationary moving-average process having the representation where the real-valued process (Yn) has a well-defined entropy and spectrum. Let ∊^∗2k denote the smallest mean-squared error of any estimate of Xn based on observations of Xn–1, Xn–2, …, Xn–k, and let ∊^∗2klin, 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 {Yn}. We also obtain explicit formulas for ∊^∗2k and ∊^∗2klin and compare these quantities graphically when M = 2 and the {Yn} 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.