Annals of Statistics

Hurst function estimation

Jinqi Shen and Tailen Hsing

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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.

Article information

Ann. Statist., Volume 48, Number 2 (2020), 838-862.

Received: June 2018
Revised: January 2019
First available in Project Euclid: 26 May 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 62M30: Spatial processes

Infill asymptotics minimax rate multifractional Brownian motion nonparametric estimation


Shen, Jinqi; Hsing, Tailen. Hurst function estimation. Ann. Statist. 48 (2020), no. 2, 838--862. doi:10.1214/19-AOS1825.

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  • [1] Bardet, J.-M. and Surgailis, D. (2013). Nonparametric estimation of the local Hurst function of multifractional Gaussian processes. Stochastic Process. Appl. 123 1004–1045.
  • [2] Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoam. 13 19–90.
  • [3] Bertrand, P. R., Fhima, M. and Guillin, A. (2013). Local estimation of the Hurst index of multifractional Brownian motion by increment ratio statistic method. ESAIM Probab. Stat. 17 307–327.
  • [4] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
  • [5] Coeurjolly, J.-F. (2005). Identification of multifractional Brownian motion. Bernoulli 11 987–1008.
  • [6] Cohen, S. (1999). From self-similarity to local self-similarity: The estimation problem. In Fractals: Theory and Applications in Engineering 3–16. Springer, London.
  • [7] Falconer, K. J. (2002). Tangent fields and the local structure of random fields. J. Theoret. Probab. 15 731–750.
  • [8] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. CRC Press, London.
  • [9] Herbin, E. (2006). From $N$ parameter fractional Brownian motions to $N$ parameter multifractional Brownian motions. Rocky Mountain J. Math. 36 1249–1284.
  • [10] Hsing, T., Brown, T. and Thelen, B. (2016). Local intrinsic stationarity and its inference. Ann. Statist. 44 2058–2088.
  • [11] Lévy-Véhel, J. and Peltier, R. F. (1995). Multifractional Brownian motion: Definition and preliminary results. Rapport de Recherche de L’INRIA N2645.
  • [12] Loh, W.-L. (2015). Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations. Ann. Statist. 43 2766–2794.
  • [13] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [14] Nourdin, I. (2012). Selected Aspects of Fractional Brownian Motion. Bocconi & Springer Series 4. Springer, Milan.
  • [15] Shen, J. and Hsing, T. (2020). Supplement to “Hurst function estimation.”
  • [16] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics. Springer, New York.
  • [17] Stoev, S. A. and Taqqu, M. S. (2006). How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 200–221.
  • [18] Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • [19] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York. Revised and extended from the 2004 French original. Translated by Vladimir Zaiats.

Supplemental materials

  • Supplement to “Hurst function estimation”. This supplemental material includes all the proofs not included in the paper.