We estimate the Hurst parameter H of a fractional Brownian motion from discrete noisy data observed along a high frequency sampling scheme. The presence of systematic experimental noise makes recovery of H more difficult since relevant information is mostly contained in the high frequencies of the signal.
We quantify the difficulty of the statistical problem in a min-max sense: we prove that the rate n−1/(4H+2) is optimal for estimating H and propose rate optimal estimators based on adaptive estimation of quadratic functionals.
"Estimation of the Hurst parameter from discrete noisy data." Ann. Statist. 35 (5) 1947 - 1974, October 2007. https://doi.org/10.1214/009053607000000316