Nonparametric estimation of the local Hurst function of multifractional Gaussian processes

TL;DR

This paper introduces a new nonparametric estimator for the local Hurst function of multifractional Gaussian processes, demonstrating its consistency, asymptotic properties, and superior accuracy over existing methods through simulations.

Contribution

It proposes a novel IR-based estimator for the local Hurst function with proven consistency, asymptotic normality, and improved accuracy compared to the QV estimator.

Findings

IR-estimator outperforms QV-estimator in accuracy

Establishes consistency and CLT for the estimators

Detailed analysis of multifractional Brownian motion

Abstract

A new nonparametric estimator of the local Hurst function of a multifractional Gaussian process based on the increment ratio (IR) statistic is defined. In a general frame, the point-wise and uniform weak and strong consistency and a multidimensional central limit theorem for this estimator are established. Similar results are obtained for a refinement of the generalized quadratic variations (QV) estimator. The example of the multifractional Brownian motion is studied in detail. A simulation study is included showing that the IR-estimator is more accurate than the QV-estimator.