Wavelet eigenvalue regression for [formula omitted]-variate operator fractional Brownian motion

Abstract

In this paper, we extend the methodology proposed in Abry and Didier (2018) to obtain the first joint estimator of the real parts of the Hurst eigenvalues of n-variate operator fractional Brownian motion (OFBM). The procedure consists of a weighted 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 (entry-wise) 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.