Robust estimation of stationary continuous-time ARMA models via indirect inference

TL;DR

This paper introduces a robust indirect estimation method for continuous-time ARMA models that remains effective under non-Gaussian noise and outliers, combining GM-estimators and simulation-based parameter matching.

Contribution

It develops a new robust estimation procedure for CARMA models using indirect inference with GM-estimators, addressing non-Gaussian noise and outliers.

Findings

The GM-estimator for AR(r) parameters is consistent and asymptotically normal.

The indirect estimator for CARMA parameters is consistent and asymptotically normal.

The method demonstrates robustness against outliers in simulation studies.

Abstract

In this paper we present a robust estimator for the parameters of a continuous-time ARMA(p,q) (CARMA(p,q)) process sampled equidistantly which is not necessarily Gaussian. Therefore, an indirect estimation procedure is used. It is an indirect estimation because we first estimate the parameters of the auxiliary AR(r) representation ($r\geq 2p-1$) 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 parameter which takes the minimum distance between the estimated AR parameters and the simulated AR parameters gives 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. Next, we prove that the indirect estimator is consistent and asymptotically normally distributed as well using in the simulation part the asymptotically normally distributed LS-estimator. The indirect estimator satisfies several important robustness properties such as weak resistance, $\pi_{d_n}$-robustness and it has a bounded influence functional. The practical applicability of our method is demonstrated through a simulation study with replacement outliers and compared to the non-robust quasi-maximum-likelihood estimation method.