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Nonparametric estimation of the local Hurst function of multifractional Gaussian processes

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  • Bardet, Jean-Marc
  • Surgailis, Donatas

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.

Suggested Citation

  • Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Nonparametric estimation of the local Hurst function of multifractional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1004-1045.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:1004-1045
    DOI: 10.1016/j.spa.2012.11.009
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    References listed on IDEAS

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    1. Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Moment bounds and central limit theorems for Gaussian subordinated arrays," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 457-473.
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    6. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    7. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
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    Cited by:

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    2. Ran Wang & Yimin Xiao, 2022. "Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2442-2479, December.
    3. Matthieu Garcin, 2019. "Hurst Exponents And Delampertized Fractional Brownian Motions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(05), pages 1-26, August.
    4. Joachim Lebovits & Mark Podolskij, 2016. "Estimation of the global regularity of a multifractional Brownian motion," CREATES Research Papers 2016-33, Department of Economics and Business Economics, Aarhus University.
    5. Li, Yuan & Pakkanen, Mikko S. & Veraart, Almut E.D., 2023. "Limit theorems for the realised semicovariances of multivariate Brownian semistationary processes," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 202-231.
    6. Sixian Jin & Qidi Peng & Henry Schellhorn, 2018. "Estimation of the pointwise Hölder exponent of hidden multifractional Brownian motion using wavelet coefficients," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 113-140, April.
    7. Mathias Mørck Ljungdahl & Mark Podolskij, 2020. "A minimal contrast estimator for the linear fractional stable motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 381-413, July.
    8. Antoine Echelard & Jacques Lévy Véhel & Anne Philippe, 2015. "Statistical Estimation for a Class of Self-Regulating Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 485-503, June.
    9. Matthieu Garcin, 2018. "Hurst exponents and delampertized fractional Brownian motions," Working Papers hal-01919754, HAL.
    10. Mazur, Stepan & Otryakhin, Dmitry & Podolskij, Mark, 2018. "Estimation of the linear fractional stable motion," Working Papers 2018:3, Örebro University, School of Business.
    11. Loboda, Dennis & Mies, Fabian & Steland, Ansgar, 2021. "Regularity of multifractional moving average processes with random Hurst exponent," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 21-48.
    12. Peng, Qidi & Zhao, Ran, 2018. "A general class of multifractional processes and stock price informativeness," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 248-267.
    13. Bianchi, Sergio & Pianese, Augusto, 2018. "Time-varying Hurst–Hölder exponents and the dynamics of (in)efficiency in stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 64-75.

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