Fast and Robust Parametric Estimation for Time Series and Spatial Models

Abstract

We present a new framework for robust estimation and inference on second-order stationary time series and random fields. This framework is based on the Generalized Method of Wavelet Moments which uses the wavelet variance to achieve parameter estimation for complex models. Using an M-estimator of the wavelet variance, this method can be made robust therefore allowing to estimate the parameters of a wide range of time series and spatial models when the data suffers from outliers or different forms of contamination. The paper presents a series of simulation studies as well as a range of applications where this new approach can be considered as a computationally efficient, numerically stable and robust method which performs at least as well as existing methods in bounding the influence of outliers on the estimation procedure.

Wavelet-Based Moment-Matching Techniques for Inertial Sensor Calibration

“…Finally, Assumption G is required for any estimator which makes use of moments (such as the AV or WV) to deliver asymptotic normality of the estimator itself. This assumption is verified under few additional conditions compared to those required for Assumption C, as highlighted again in [11], [14] and, under weaker conditions, in [12]. Using these assumptions, we obtain the following result.…”

“…Its main purpose is to ensure that certain quantities that we will consider in the proofs will be bounded in order to ensure convergence. Assumption C is rather mild and lower-level conditions equivalent to this assumption can, for example, be found in [11] for the WV (as well as in [12] under weaker conditions) or by combining these results with the work of [7] who showed the equivalence between the AV and WV. Finally, Assumption D requires the function f (ν(θ)) to be continuous in Θ which is the case when both f (•) and ν(•) are continuous within their respective composition domains.…”

“…As shown in the corollary below, the optimal choice (in terms of asymptotic efficiency) of Ω * is the inverse of F(θ 0 ) V(θ 0 ). Although the true V(θ 0 ) is unknown in practice, it can be consistently estimated by the estimator proposed in [15] or with the approach discussed in [12]. Moreover, the corollary of Theorem 2 presented below shows how asymptotically optimal estimators can be constructed for the GMWFM.…”