Wavelet-Variance-Based Estimation for Composite Stochastic Processes

Abstract

This article presents a new estimation method for the parameters of a time series model. 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 estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator furnishes 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 inference and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also set sufficient conditions on composite models for our estimator to be consistent, that are easy to verify. 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. Supplementary materials for this article are available online.

Estimation of Time Series Models via Robust Wavelet Variance

“…The theoretical WV of many processes can be derived from the results in Zhang (2008) or, as an alternative, Guerrier et al (2013b) suggest to use indirect inference to overcome the complexity of calculations for certain models. Hence, the proposed estimator is easily implemented and computationally inexpensive while at the same time providing a robust estimation method for many processes for whom robust estimation methods are scarce.…”

“…, ν J ] as the vector of WV and ν(θ) as the WV vector implied by the process P θ . Taking advantage of the above link, Guerrier et al (2013b) propose the following estimatorθ = argmin…”

“…For these kind of processes, the GMWM (along with the robust version presented in this paper) presents important advantages over alternative approaches (see Guerrier et al (2013b)). When contaminating this process (with φ = 0.95, ω = 0.04, σ 2 AR = 16, σ 2 W N = 4) with 5%-additive outliers with σ 2 = 9, the results seem to indicate that the classic GMWM does not appear to be greatly affected by this contamination for the first three parameters.…”

“…An example of such a process was given in Guerrier et al (2013b) as a sum MLE GMWM RGMWM 1.0 1.2 1.4 1.6 1.8 Figure 2: Finite sample performance of the MLE, GMWM and RGMWM estimators on a 5%-contaminated white noise process of length L = 1, 000, with σ = 1 and contamination generated by adding Gaussian noise with σ 2 = 100. MLE represents the maximum likelihood estimator, GMWM represents the classic GMWM estimator with additional second moment of the process, RGMWM represents the robust GMWM based on the M-estimator proposed by Mondal and Percival (2012) with DWT wavelet transforms.…”

“…We will investigate the performance of the estimator on three processes, namely a white noise process (WN), a first-order autoregressive process (AR1) and a composite stochastic process like the simulation presented in Guerrier et al (2013b).…”