  Complex time series models such as (the sum of) ARMA$(p,q)$ models with
additional noise, random walks, rounding errors and/or drifts are increasingly
used for data analysis in fields such as biology, ecology, engineering and
economics where the length of the observed signals can be extremely large.
Performing inference on and/or prediction from these models can be highly
challenging for several reasons: (i) the data may contain outliers that can
adversely affect the estimation procedure; (ii) the computational complexity
can become prohibitive when models include more than just a few parameters
and/or the time series are large; (iii) model building and/or selection adds
another layer of (computational) complexity to the previous task; and (iv)
solutions that address (i), (ii) and (iii) simultaneously do not exist in
practice. For this reason, this paper aims at jointly addressing these
challenges by proposing a general framework for robust two-step estimation
based on a bounded influence M-estimator of the wavelet variance. In this
perspective, we first develop the conditions for the joint asymptotic normality
of the latter estimator thereby providing the necessary tools to perform
(direct) inference for scale-based analysis of signals. Taking advantage of the
model-independent weights of this first-step estimator that are computed only
once, we then develop the asymptotic properties of two-step robust estimators
using the framework of the Generalized Method of Wavelet Moments (GMWM), hence
defining the Robust GMWM (RGMWM) that we then use for robust model estimation
and inference in a computationally efficient manner even for large time series.
Simulation studies illustrate the good finite sample performance of the RGMWM
estimator and applied examples highlight the practical relevance of the
proposed approach.
