Eneralized method of wavelet moments for the estimation of composite stochastic models

Abstract

We present a new estimation method for the parameters of a times series model called the Generalised Method of Wavelet Moments (GMWM) estima- tor; see Guerrier et al. (2013). We consider here composite Gaussian processes that are the sum of independent Gaussian processes which in turn explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical esti- mation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for infer- ence and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also propose a robust alternative, based on a robust WV estimator. We use the new estimator to estimate the stochastic error’s parameters of the sum of three first order Gauss-Markov processes by means of a sample of over 800,000 issued from gyroscopes that compose inertial navigation systems.