On Maximum Likelihood Estimation of Parameters of Autoregressive Moving Average Processes

Abstract

A stationary stochastic process that serves as a useful model for time series analysis is the autoregressive process with moving average residuals {y _t } which satisfies

$$\sum\limits_{j = 0}^p {{\beta _j}{y_{t - j}}} = \sum\limits_{g = 0}^q {{\alpha _g}{v_{t - g}}} $$

((1))

, t = ..., - 1, 0, 1,..., where the sequence {v _t } consists of independently identically distributed (unobservable) random variables. To avoid indeterminancy β ₀ = α ₀ = 1. The mean of v _t is independent of t and is taken to be 0 for convenience; the variance of v _t is σ ². We shall assume that the v _t ’s are normally distributed, that is, that the process is Gaussian.

Research supported by the U.S. Office of Naval Research under Contract Number N00014-67-A-0112-0030. The author thanks Paul Shaman for helpful discussions.