Comments on Estimation and Prediction for Autoregressive and Moving Average Nongaussian Sequences

Abstract

Autoregressive and moving average models have been studied for a long period of time, particularly in the Gaussian case, with respect to the problems of prediction and that of estimation of the coefficients of the models. It is only in recent years that especial attention has been paid to the case of nonGaussian models where it has been realized that the corresponding problems may have a more complicated but richer structure. A discrete time autoregressive moving average model is a solution x _t of the system of equations

$$ \sum\limits_{j=0}^p{{a_j}{x_{t-j}}}=\sum\limits_{k=0}^q {{b_k}{\xi_{t-k}}}$$

(1.1)

where the sequence ξ _t is a sequence of independent, identically distributed random variables with Eξ _t ≡ 0, Eξ ² _t = σ ² > 0(σ ² < ∞). The coefficients a _j , b _k are real and it is conventional to set a ₀ = b ₀ = 1. There is a stationary solution to the system if and only if the polynomial

$$ a\left(z\right)=\sum\limits_{j=0}^p{{a_j}{z^j}} $$

has no zeros of absolute value one and this solution is uniquely determined. The estimation problem is that of estimating the coefficients a _j , b _k given the sequence of observations x ₁,…, x _n . The stationary solution x _t is causal, if the polynomial a(z) has all its zeros of absolute value greater than one, in the sense that there is a one-sided representation of x _t in terms of the present and past of ξ the sequence

$$ {x_t}=\sum\limits_{j=0}^\infty{{\alpha_j}{\xi_{t-j}}} $$

with the coefficients α _j decreasing to zero exponentially fast as j → ∞.

This research was supported in part by Office of Naval Research Grant N00014-90-J1372.