## The Annals of Statistics

### Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains

T. W. Anderson

#### Abstract

The autoregressive moving average model is a stationary stochastic process $\{y_t\}$ satisfying $\sum^p_{k=0} \beta_ky_{t-k} = \sum^q_{g=0} \alpha_g\nu_{t-g}$, where the (unobservable) process $\{v_t\}$ consists of independently identically distributed random variables. The coefficients in this equation and the variance of $v_t$ are to be estimated from an observed sequence $y_1, \cdots, y_T$. To apply the method of maximum likelihood under normality the model is modified (i) by setting $y_0 = y_{-1} = \cdots = y_{1-p} = 0$ and $\nu_0 = v_{-1} = \cdots = v_{1-q} = 0$ and alternatively (ii) by setting $y_0 \equiv y_T, \cdots, y_{1-p} \equiv y_{T+1-p}$ and $v_0 \equiv v_T, \cdots, v_{1-q} \equiv v_{T+1-q}$; the former lead to procedures in the time domain and the latter to procedures in the frequency domain. Matrix methods are used for a unified development of the Newton-Raphson and scoring iterative procedures; most of the procedures have been obtained previously by different methods. Estimation of the covariances of the moving average part is also treated.

#### Article information

Source
Ann. Statist., Volume 5, Number 5 (1977), 842-865.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176343942

Digital Object Identifier
doi:10.1214/aos/1176343942

Mathematical Reviews number (MathSciNet)
MR448762

Zentralblatt MATH identifier
0368.62075

JSTOR