Robust Two-Step Wavelet-Based Inference for Time Series Models

Abstract

Latent time series models such as (the independent sum of ) ARMA(p, q) models with additional stochastic processes are increasingly used for data analysis in biology, ecology, engineering, and economics. Inference on and/or prediction from these models can be highly challenging: (i) the data may contain outliers that can adversely affect the estimation procedure; (ii) the computational complexity can become prohibitive when the time series are extremely large; (iii) model selection adds another layer of (computational) complexity; and (iv) solutions that address (i), (ii), and (iii) simultaneously do not exist in practice. 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. 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, we then develop the asymptotic properties of two-step robust estimators using the framework of the generalized method of wavelet moments (GMWM). Simulation studies illustrate the good finite sample performance of the robust GMWM estimator and applied examples highlight the practical relevance of the proposed approach.

Scale-wise Variance Minimization for Optimal Virtual Signals: An Approach for Redundant Gyroscopes

“…In particular, Condition (C3) ensures that the cumulative impact of * 0 on the future values of the process (W i,j,t ) is finite, and therefore, it can be interpreted as a short-range dependence condition [25]. REMARK B: As underlined in [26], when using a Daubechies filter (such as the Haar wavelet filter) the wavelet coefficients (W i,j,t ) can be represented as a linear combination of the dth order difference 1 of the original process (X i;t ), which we denote as (∆ i;t ). In this case Conditions (C1) to (C3) can be directly applied to (∆ i;t ) instead of on (W i,j,t ).…”

“…Among others, examples of such applications are related to Portmanteau tests, model estimation and selection (see e.g. [2,26,33,34,35,36,37] to mention a few). This approach is consequently used for the simulation and case studies presented in the following sections also highlighting its relevance in the context of this work.…”

Multi-Signal Approaches for Repeated Sampling Schemes in Inertial Sensor Calibration

“…More specifically, in the following sections, we describe and study the solutions, including those put forward in [9] and [10], which are a direct extension of the GMWM. The latter is currently employed, among others, for sensor calibration on a single stochastic error signal issued from an inertial sensor calibration session (see, e.g., [8], [11]). Indeed, in order to estimate the parameter vector (θ ∈ IR p ) that characterizes the model underlying the stochastic error, denoted as F θ , the GMWM is defined as follows:…”

“…where, with Z ∈ IR J , we have that Z 2 Ω := Z ΩZ. In addition, ν represents the WV estimated on the single error signal issued from the calibration session, ν(θ) represents the theoretical WV implied by the parametric model F θ and Ω is a positive definite weighting matrix chosen in a suitable way (see, e.g., [11] and following sections for more details).…”

“…Finally, Assumption C requires consistency of the WV estimator (which was proven under different conditions, see e.g. [11]) as well as that of Ω (if an estimator is actually chosen for the weighting matrix Ω).…”