Wavelet eigenvalue regression for $n$-variate operator fractional Brownian motion

TL;DR

This paper introduces a wavelet eigenvalue regression method to estimate the Hurst eigenvalues of multivariate operator fractional Brownian motion, demonstrating its consistency, asymptotic normality, and effectiveness through simulations and real Internet traffic data analysis.

Contribution

It extends previous methods to jointly estimate Hurst eigenvalues in multivariate OFBM, providing a consistent and asymptotically normal estimator with practical advantages.

Findings

Estimator is consistent for any time reversible OFBM.

Simulation studies show the method's finite sample effectiveness.

Application reveals complex multivariate self-similarity in Internet traffic.

Abstract

In this contribution, we extend the methodology proposed in Abry and Didier (2017) 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.