Robust estimation of stationary continuous‐time arma models via indirect inference

Abstract

In this article, we present a robust estimator for the parameters of a stationary, but not necessarily Gaussian, continuous‐time ARMA(p,q) (CARMA(p,q)) process that is sampled equidistantly. Therefore, we propose an indirect estimation procedure that first estimates the parameters of the auxiliary AR(r) representation (r ≥ 2p−1) of the sampled CARMA process using a generalized M‐ (GM‐)estimator. Since the map which maps the parameters of the auxiliary AR(r) representation to the parameters of the CARMA process is not given explicitly, a separate simulation part is necessary where the parameters of the AR(r) representation are estimated from simulated CARMA processes. Then, the parameters which take the minimum distance between the estimated AR parameters and the simulated AR parameters give an estimator for the CARMA parameters. First, we show that under some standard assumptions the GM‐estimator for the AR(r) parameters is consistent and asymptotically normally distributed. Then, we prove that the indirect estimator is also consistent and asymptotically normally distributed when the asymptotically normally distributed LS‐estimator is used in the simulation part. The indirect estimator satisfies several important robustness properties such as weak resistance, ‐robustness and it has a bounded influence functional. The practical applicability of our method is illustrated in a small simulation study with replacement outliers.