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Testing of fractional Brownian motion in a noisy environment

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  • Balcerek, Michał
  • Burnecki, Krzysztof

Abstract

Fractional Brownian motion (FBM) is related to the notions of self-similarity, ergodicity and long memory. These properties have made FBM important in modeling real-world phenomena in different experiments ranging from telecommunication to biology. However, these experiments are often disturbed by a noise which source can be, e.g., the instrument error. In this paper we propose a rigorous statistical test for FBM with added white Gaussian noise which is based on the autocovariance function. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power for alternatives being FBM’s with different self-similarity parameters and the scaled Brownian motion which is also Gaussian and self-similar. We note that the introduced test can be adapted to an arbitrary Gaussian process with a given covariance structure.

Suggested Citation

  • Balcerek, Michał & Burnecki, Krzysztof, 2020. "Testing of fractional Brownian motion in a noisy environment," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    • Handle: RePEc:eee:chsofr:v:140:y:2020:i:c:s096007792030494x
    DOI: 10.1016/j.chaos.2020.110097
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    References listed on IDEAS

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    1. Pawe³ Bieñkowski & Krzysztof Burnecki & Joanna Janczura & Rafal Weron & Bart³omiej Zubrzak, 2012. "A new method for automated noise cancellation in electromagnetic field measurement," HSC Research Reports HSC/12/05, Hugo Steinhaus Center, Wroclaw University of Technology.
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    3. Burnecki, Krzysztof & Gajda, Janusz & Sikora, Grzegorz, 2011. "Stability and lack of memory of the returns of the Hang Seng index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(18), pages 3136-3146.
    4. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    5. Sikora, Grzegorz, 2018. "Statistical test for fractional Brownian motion based on detrending moving average algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 54-62.
    6. Rafal Weron, 2006. "Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook0601.
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    2. Vasile Brătian & Ana-Maria Acu & Camelia Oprean-Stan & Emil Dinga & Gabriela-Mariana Ionescu, 2021. "Efficient or Fractal Market Hypothesis? A Stock Indexes Modelling Using Geometric Brownian Motion and Geometric Fractional Brownian Motion," Mathematics, MDPI, vol. 9(22), pages 1-20, November.

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