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

Abstract

Self-similarity has been widely used to model scale-free dynamics, with significant successes in numerous applications that are very different in nature. However, such successes have mostly remained confined to univariate data analysis while many applications in the modern “data deluge” era involve multivariate and dependent data. Operator fractional Brownian motion is a multivariate self-similar model that accounts for multivariate scale-free dynamics and characterizes data by means of a vector of self-similarity exponents (eigenvalues). This naturally raises the challenging question of testing the equality of exponents. Expanding on the recently proposed wavelet eigenvalue regression estimator of the vector of self-similarity exponents, in the present work we construct and study a wavelet domain bootstrap test for the equality of self-similarity exponents from one single observation (time series) of multivariate data. Its performance is assessed in a bivariate setting for various choices of sample size and model parameters, and it is shown to be satisfactory for use on real world data. Practical routines implementing estimation and testing are available upon request.