Influence Functionals for Time Series

Abstract

A definition is given for influence functionals of parameter estimates in time-series models. The definition involves the use of a contaminated observations process of the form $y^\gamma_t=(1-z^\gamma_t){x_t+z^\gamma_tw_t}$, $p=1,2,...,0\leq\gamma\leq1$ where $x_t$ is a core process (usually Gaussian), $w_t$ is a contaminating process, and $z^\gamma_t"$ is a zero-one process with $P(z^\gamma_t=1)={\gamma+0(\gamma)}$. This form is sufficiently general to model such diverse contamination types as isolated outliers and patches of outliers. Let $T(\mu^\gamma_y)$ denote the functional representation of a given estimate, where the measures $\mu^\gamma_y, 0\leq\gamma\leq1$ for $y^\gamma_t$ are in an appropriate subset of the family of stationary and ergodic measures on $(R^\infty,\beta^\infty)$. The influence functional IF is a derivative of T along "arcs" traced by $\mu^\gamma_y$ as $\gamma\rigtharrow0$, and correspondingly $\mu^\gamma_y\rigtharrow\gamma_x$. Although this influence functional is similar in spirit to Hampel's influence curve ICH for the i. i.d. setting, it is not the same as ICH. However, a simple relationship between the IF and the ICH is established. Results are given which aid in the computation of IF and insure that IF is bounded. We compute the IF for some robust estimates of the first-order autoregressive and first-order moving average parameters using various contamination processes. A definition of gross-error sensitivity (GES) for the IF is given, and some estimates are compared in terms of their GES's. Also the IF is used to show that a class of generalized RA estimates has a certain optimality property. Finally, some possible generalizations of the IF are indicated.