Wavelet eigenvalue regression in high dimensions

Patrice Abry, B. Cooper Boniece (), Gustavo Didier and Herwig Wendt
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Patrice Abry: Université de Lyon
B. Cooper Boniece: University of Utah
Gustavo Didier: Tulane University
Herwig Wendt: Université de Toulouse

Statistical Inference for Stochastic Processes, 2023, vol. 26, issue 1, No 1, 32 pages

Abstract: Abstract In this paper, we construct the wavelet eigenvalue regression methodology (Abry and Didier in J Multivar Anal 168:75–104, 2018a; in Bernoulli 24(2):895–928, 2018b) in high dimensions. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate ( $$r \ll p$$ r ≪ p ) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension r of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations.

Keywords: Wavelets; Operator self-similarity; Random matrices; Primary 62H25; 60B20; Secondary 42C40; 60G18 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s11203-022-09279-3

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