On Prediction of Moving-Average Processes

This paper is concerned with the problem of estimating the current value, Xn, of a discrete-time stationary stochastic process given the k previous values, X,,-u Xn-2, · · ·, Xn-k. Denote such an estimator, or predictor, by = fk(Xn-1, Xn-2, We adopt the mean-squared error el = E[Xn - X ] 2 (2) (1) as a figure of merit for the estimator fk. Throughout the paper, we assume EXn = 0, n = 0, ±1, ±2, . . . . It is well known, and easy to show, that no estimator has smaller mean-squared error than fk(Xn-1, . · ., Xn-k) = E{Xn Xn-l, ' ' Xn-k), (3) the conditional expectation of Xn given Xn-, · · *, Xn-k. We denote the 367 mean-squared error of this best estimator by t2 = E[Xn - ft]2 = E[Xn - E(XnXn-l, Xn-k)f. (4) While the best estimator is simply described by (3), in practice it is frequently impossible to calculate it explicitly for processes of interest. The simpler class of linear estimators k Alin(Xi-l, ' ' '» Xn-k) = X CkjXn-j (5) 1 has been much studied in the past.1 It is well known how to choose the c's to obtain the smallest mean-squared error within this class of predictors, and this least mean-squared error is given by the simple formula jfl?n = Dk+i/Dk, (6)