This paper considers a wide range of issues concerning the estimation of the Hurst function of a multifractional Brownian motion when the process is observed on a regular grid. A theoretical lower bound for the minimax risk of this inference problem is established for a wide class of smooth Hurst functions. We also propose a new nonparametric estimator and show that it is rate optimal. Implementation issues of the estimator including how to overcome the presence of a nuisance parameter and choose the tuning parameter from data will be considered. An extensive numerical study is conducted to compare our approach with other approaches.
"Hurst function estimation." Ann. Statist. 48 (2) 838 - 862, April 2020. https://doi.org/10.1214/19-AOS1825