Wavelet eigenvalue regression for n-variate operator fractional Brownian motion

Abstract

In this contribution, we extend the methodology proposed in Abry and Didier (2017a) to obtain the first joint estimator of the real parts of the Hurst eigenvalues of n-variate OFBM. The procedure consists of a wavelet regression on the log-eigenvalues of the sample wavelet spectrum. The estimator is shown to be consistent for any time reversible OFBM and, under stronger assumptions, also asymptotically normal starting from either continuous or discrete time measurements. Simulation studies establish the finite sample effectiveness of the methodology and illustrate its benefits compared to univariate-like (entrywise) analysis. As an application, we revisit the well-known self-similar character of Internet traffic by applying the proposed methodology to 4-variate time series of modern, high quality Internet traffic data. The analysis reveals the presence of a rich multivariate self-similarity structure.

Bootstrap-based Bias Reduction for the Estimation of the Self-similarity Exponents of Multivariate Time Series

“…However, in the general case when components are mixtures of fBms, i.e., when W is non-diagonal, univariatelike estimation of H is, in general, strongly biased. In [12,14], a wavelet-based multivariate statistical strategy is proposed. The multivariate discrete wavelet transform coefficients are computed as [20,21].…”

“…Details on the numerical simulation are given in Section 4 below. In light of the proven consistency of wavelet log-eigenvalues for Hurst exponents [12,14], the naive expectation would be to observe nearly parallel straight wavelet log-eigenvalue lines as the octave j increases. However, what the plots show is rather different.…”

“…, M . OfBm has now been successfully applied in many fields such as in Internet traffic modeling [12] and dendrochronology [13].…”

“…For ofBm, univariate-like statistical methods such as the entrywise analysis of sample covariance matrices, Fourier or wavelet spectra does not yield relevant results. In [12,14], a wavelet eigenvalue regression methodology is constructed for the joint estimation of the vector of ordered Hurst exponents H = (H1, . .…”

Wavelet Domain Bootstrap for Testing the Equality of Bivariate Self-Similarity Exponents

“…Self-similarity [1] provides a framework for describing and modeling scale-free dynamics. It has been widely used and lead to well-recognized successes in numerous real world applications that are very different in nature (cf., e.g., [2][3][4] and references therein). Fractional Brownian motion (fBm) is the only Gaussian stationary increment self-similar process [5].…”