An Overview of Robust Spectral Estimators

Abstract

The periodogram function is widely used to estimate the spectral density of time series processes and it is well-known that this function is also very sensitive to outliers. In this context, this paper deals with robust estimation functions to estimate the spectral density of univariate and periodic time series with short and long-memory properties. The two robust periodogram functions discussed and compared here were previously explicitly and analytically derived in Fajardo et al. (2018), Reisen et al. (2017) and Fajardo et al. (2009) in the case of long-memory processes. The first two references introduce the robust periodogram based on M-regression estimator. The third reference is based on the robust autocovariance function introduced in Ma and Genton (2000) and studied theoretically and empirically in Lévy-Leduc et al. (2011). Here, the theoretical results of these estimators are discussed in the case of short and long-memory univariate time series and periodic processes. A special attention is given to the M-periodogram for short-memory processes. In this case, Theorem 1 and Corollary 1 derive the asymptotic distribution of this spectral estimator. As the application of the methodologies, robust estimators for the parameters of AR, ARFIMA and PARMA processes are discussed. Their finite sample size properties are addressed and compared in the context of absence and presence of atypical observations. Therefore, the contributions of this paper come to fill some gaps in the literature of modeling univariate and periodic time series to handle additive outliers.