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, d n -robustness and it has a bounded influence functional. The practical applicability of our method is illustrated in a small simulation study with replacement outliers.

Multivariate continuous-time ARMA(p, q) (MCARMA(p, q)) processes are the continuous-time analog of the well-known vector ARMA(p, q) processes. They have attracted interest over the last years. Methods to estimate the parameters of an MCARMA process require an identifiable parametrization such as the Echelon form with a fixed Kronecker index, which is in the one-dimensional case the degree p of the autoregressive polynomial. Thus, the Kronecker index has to be known in advance before the parameter estimation is done. When this is not the case information criteria can be used to estimate the Kronecker index and the degrees (p, q), respectively. In this paper we investigate information criteria for MCARMA processes based on quasi maximum likelihood estimation. Therefore, we first derive the asymptotic properties of quasi maximum likelihood estimators for MCARMA processes in a misspecified parameter space. Then, we present necessary and sufficient conditions for information criteria to be strongly and weakly consistent, respectively. In particular, we study the well-known Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as special cases.

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In this paper we present a robust estimator for the parameters of a stationary continuoustime 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 ≥ 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 GMestimator 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 LSestimator. The indirect estimator satisfies several important robustness properties such as weak resistance, π 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.