Jean-Marc Bardet

Wavelet Estimator of Long-Range Dependent Processes

In this contribution, the statistical properties of the wavelet estimator of the long-range dependence parameter introduced in Abry et al. (1995) are discussed for a stationary Gaussian process. This contribution complements the heuristical discussion presented in Abry et al. (1999), by taking into account the correlation between the wavelet coefficients (which is discarded in the mentioned reference) and the bias due to the short-memory component. We derive expressions for the estimators asymptotic bias, variance and mean-squared error as functions of the scale used in the regression and some user-defined parameters. Consistency of the estimator is obtained as long as the scale index jT goes to infinity and 2jT/T→0, where T denotes the sample size. Under these and some additional conditions assumed in the paper, we also establish the asymptotic normality of this estimator.

ASYMPTOTIC NORMALITY OF THE QUASI MAXIMUM LIKELIHOOD ESTIMATOR FOR MULTIDIMENSIONAL CAUSAL PROCESSES

Strong consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) are given for a general class of multidimensional causal processes. For particular cases already studied in the literature (for instance univariate or multivariate GARCH, ARCH, ARMA-GARCH processes) the assumptions required for establishing these results are often weaker than existing conditions. The QMLE asymptotic behavior is also given for numerous new examples of univariate or multivariate processes (for instance TARCH or NLARCH processes).

Statistical study of the wavelet analysis of fractional Brownian motion

We present a statistical study of wavelet coefficients of a fractional Brownian motion. A central limit theorem for empirical variances of exact wavelet coefficients is given. Under conditions on the mother wavelet and the choice of scales, a limit theorem is given for fitted wavelet coefficients computed from a time series. It provides an estimator for the self-similarity parameter of Gaussian time series.

Asymptotic Properties of the Detrended Fluctuation Analysis of Long-Range-Dependent Processes

In the past few years, a certain number of authors have proposed analysis methods of the time series built from a long-range dependence noise. One of these methods is the detrended fluctuation analysis (DFA), frequently used in the case of physiological data processing. The aim of this method is to highlight the long-range dependence of a time series with trend. In this paper, asymptotic properties of the DFA of the fractional Gaussian noise (FGN) are provided. Those results are also extended to a general class of stationary long-range-dependent processes. As a consequence, the convergence of the semiparametric estimator of the Hurst parameter is established. However, several simple examples also show that this method is not at all robust in the case of trends.

Identification of the multiscale fractional Brownian motion with biomechanical applications

In certain applications, for instance, biomechanics, turbulence, finance or internet traffic, it seems suitable to model the data by a generalization of a fractional Brownian motion (FBM) for which the Hurst parameter H depends on the frequency as a piece-wise constant function. These processes are called multiscale fractional Brownian motions. In this article, we provide a statistical study of the multiscale fractional Brownian motions. We developed a method based on wavelet analysis. By using this method, we calculated the frequency changes, estimated the different parameters, tested the goodness-of-fit and gave the numerical algorithm. Biomechanical data are then studied with these new tools. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.

Testing for the Presence of Self-Similarity of Gaussian Time Series Having Stationary Increments

A method for testing for the presence of self-similarity of a Gaussian time series with stationary increments is presented. The test is based on estimation of the distance between the time series and a set of time series containing all the fractional Brownian motions. This distance is constructed from two estimations of multiscale generalized quadratic variations expectations. The second one requires regression estimates of the self-similarity index H. Two estimations of H are then introduced. They present good robustness and computing time properties compared with the Whittle approach, with nearly similar convergence rate. The test is applied on simulated and real data. The self-similarity assumption is notably accepted for the famous Nile River data.

A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.

Measuring the roughness of random paths by increment ratios

A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a Levy process with

DEFINITION, PROPERTIES AND WAVELET ANALYSIS OF MULTISCALE FRACTIONAL BROWNIAN MOTION

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Multiple breaks detection in general causal time series using penalized quasi-likelihood

stable tangent process is established and can be used to estimate the fractional parameter