Open Access
Translator Disclaimer
September, 1977 Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains
T. W. Anderson
Ann. Statist. 5(5): 842-865 (September, 1977). DOI: 10.1214/aos/1176343942

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.

Citation

Download Citation

T. W. Anderson. "Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains." Ann. Statist. 5 (5) 842 - 865, September, 1977. https://doi.org/10.1214/aos/1176343942

Information

Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0368.62075
MathSciNet: MR448762
Digital Object Identifier: 10.1214/aos/1176343942

Subjects:
Primary: 62M10
Secondary: 62H99

Keywords: autoregressive moving average models , maximum likelihood estimation , Newton-Raphson and scoring iterative procedures , time and frequency domains , time series analysis

Rights: Copyright © 1977 Institute of Mathematical Statistics

JOURNAL ARTICLE
24 PAGES


SHARE
Vol.5 • No. 5 • September, 1977
Back to Top