Infinitesimal Robustness for Autoregressive Processes

Abstract

We define the influence function and construct optimal robust estimators for autoregressive processes. We show that the asymptotic bias caused by small contaminations of the marginals can be written as the integral of a certain function with respect to the contamination. This function is called the influence function. It is unique only up to an equivalence relation, but there is a natural unique version which describes the limiting influence of an additional observation given the previous observations. Moreover, with this version the asymptotic variance at the true model can be expressed in a simple form. Optimal robust estimators minimize this asymptotic variance under a constraint on the influence function. As in the i.i.d case, they are found by truncating a multiple of the influence function of the maximum likelihood estimator.