Moving average processes and maximum entropy

Abstract

Summary form only given, as follows. Let Xt, t ε Z be a wide-sense stationary stochastic process with mean EXt = 0 and autocovariance γ(k) = EXt Xt+k, k ε Z. It is well known (Burg, 1967) that the maximum-entropy wide-sense stationary stochastic process that satisfies the constraints γ(i) = ci, i = 0, 1,..., p is the AR (autoregressive) Gaussian process that satisfies these constraints. Physical or practical considerations might in some cases impose the additional constraint that γ(i) = 0, i > q. Since any time series with γ(i) = 0, i > q, is an MA (moving-average) process of order (at most) q, one then faces the problem of finding the maximum-entropy process among the MA(q) processes that satisfy the constraints γ(i) = ci, i = 0,...,p. The solution to this problem rests on the relationship between the autocorrelation and inverse autocorrelation function of an AR process that was recently brought to light by Kanto (1987). It is to be noted that in the context of spectral estimation, q = p corresponds to a periodogramlike estimator, whereas q = ∞ leads to Burg's all-pole (AR) estimator. Hence the choice p < q < ∞ yields a solution intermediate between the periodogram and Burg's AR estimator.