Robust Linear Extrapolations of Second-order Stationary Processes

Abstract

This paper considers the problems of linear prediction under the condition that the spectral structures of second-order stationary processes are vaguely specified. The approach adopted is closely in line with the theory of robust estimation due to Peter Huber. The paper shows that there exists a minimax one-step ahead predictor for the set of spectral distributions given as $\{H: H = (1 - \varepsilon)F + \varepsilon G, G \in \mathscr{D}_1\}$, where $F$ is a fixed probability distribution function and $\mathscr{D}_1$ is the set of all absolutely continuous probability distribution functions. That predictor turns out to be the optimal linear predictor for a spectral distribution which is derived by a suitable modification applied to $F$. Though there generally exists no minimax predictor for the set $\{H: H = (1 - \varepsilon)F + \varepsilon G, G \in \mathscr{D}_0\}$ ($\mathscr{D}_0$ is the set of all probability distribution functions), a linear predictor is explicitly constructed so that its maximal prediction error is arbitrarily close to the lower bound of the maximal prediction errors of possible linear predictors. The results obtained in this paper would have an important application in the errors-in-variable models.