Non Gaussian autoregressive and moving average schemes. (English) Zbl 0946.62086

Rosenblatt, Murray

Szyszkowicz, Barbara (ed.), Asymptotic methods in probability and statistics. A volume in honour of Miklós Csörgő. ICAMPS ’97, an international conference at Carleton Univ., Ottawa, Ontario, Canada, July 1997. Amsterdam: North-Holland/ Elsevier. 731-737 (1998).

Introduction: Consider the system of equations \[ \sum^p_{k= 0}\widetilde\phi_k x_{t-k}= \sum^q_{j= 0}\widetilde\theta_j \xi_{t-j}, \] where \(\xi_t\)’s are independent, identically distributed random variables with \(E\xi_t\equiv 0\), \(E\xi^2_t\equiv 1\). Let \(\widetilde\phi_0= \widetilde\theta_0= 1\) and \(\widetilde\phi_p, \widetilde\theta_q\neq 0\). The polynomials \[ \phi(z)= \sum^p_{k= 0} \widetilde\phi_k z^k,\quad \theta(z)= \sum^q_{j= 0} \widetilde\theta_j z^j, \] are assumed to have no roots in common. There is then a stationary solution \(\{x_t\}\) of the system if and only if \(\phi(z)\) has no roots of absolute value one and this solution is uniquely determined. The solution is called an autoregressive moving average (ARMA) process. If \(\phi(z)\equiv 1\) the solution is a Moving Average (MA) while if \(\theta(z)\equiv 1\) it is an Autoregressive Scheme (AR). If all the roots of \(\phi(z)\) and \(\theta(z)\) are outside the unit disc \(|z|\leq 1\) in the complex plane we shall call the solution minimum phase.
It is well known that in the case of a stationary ARMA minimum phase process, the best predictor of \(x_t\) in mean square given the past is linear. A necessary condition for the best predictor to be linear is given in terms of a functional equation. This condition is then used in the case of autoregressive schemes and moving averages. A simple example is given of a nonminimum phase scheme in which the best predictor is nonlinear.
For the entire collection see [Zbl 0901.00049].

MSC:

62M20

Inference from stochastic processes and prediction