A note on estimation of $\alpha$-stable CARMA processes sampled at low frequencies

TL;DR

This paper examines the estimation of symmetric alpha-stable CARMA processes sampled at low frequencies, revealing limitations of the Whittle estimator's consistency in heavy-tailed settings and comparing it with existing estimators.

Contribution

It demonstrates that the Whittle estimator is generally inconsistent for symmetric alpha-stable CARMA processes sampled at low frequencies, contrasting with its properties in light-tailed cases.

Findings

Whittle estimator is consistent for stable CAR(1) processes.

Inconsistency of the Whittle estimator for general symmetric alpha-stable CARMA processes.

Comparison showing limitations of existing estimators in heavy-tailed settings.

Abstract

In this paper, we investigate estimators for symmetric $\alpha$-stable CARMA processes sampled equidistantly. Simulation studies suggest that the Whittle estimator and the estimator presented in Garc\'{\i}a et al. (2011) are consistent estimators for the parameters of stable CARMA processes. For CARMA processes with finite second moments it is well-known that the Whittle estimator is consistent and asymptotically normally distributed. Therefore, in the light-tailed setting the properties of the Whittle estimator for CARMA processes are similar to those of the Whittle estimator for ARMA processes. However, in the present paper we prove that, in general, the Whittle estimator for symmetric $\alpha$-stable CARMA processes sampled at low frequencies is not consistent and highlight why simulation studies suggest something else. Thus, in contrast to the light-tailed setting the properties of the Whittle estimator for heavy-tailed ARMA processes can not be transferred to heavy-tailed CARMA processes. We elaborate as well that the estimator presented in Garc\'{\i}a et al. (2011) faces the same problems. However, the Whittle estimator for stable CAR(1) processes is consistent.