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Maximum likelihood estimator for skew Brownian motion: The convergence rate

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  • Antoine Lejay
  • Sara Mazzonetto

Abstract

We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order 1/4. We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to −1 or 1.

Suggested Citation

  • Antoine Lejay & Sara Mazzonetto, 2024. "Maximum likelihood estimator for skew Brownian motion: The convergence rate," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 612-642, June.
    • Handle: RePEc:bla:scjsta:v:51:y:2024:i:2:p:612-642
    DOI: 10.1111/sjos.12694
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    References listed on IDEAS

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    1. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    2. Alexander Gairat & Vadim Shcherbakov, 2017. "Density Of Skew Brownian Motion And Its Functionals With Application In Finance," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 1069-1088, October.
    3. Said Karim Bounebache & Lorenzo Zambotti, 2014. "A Skew Stochastic Heat Equation," Journal of Theoretical Probability, Springer, vol. 27(1), pages 168-201, March.
    4. Héctor Araya & Meryem Slaoui & Soledad Torres, 2022. "Bayesian inference for fractional Oscillating Brownian motion," Computational Statistics, Springer, vol. 37(2), pages 887-907, April.
    5. Fei Su & Kung-Sik Chan, 2017. "Testing for Threshold Diffusion," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(2), pages 218-227, April.
    6. Antoine Lejay & Paolo Pigato, 2020. "Maximum likelihood drift estimation for a threshold diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 609-637, September.
    7. Marc Decamps & Marc Goovaerts & Wim Schoutens, 2006. "Self Exciting Threshold Interest Rates Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 1093-1122.
    8. Itkin, Andrey & Lipton, Alexander & Muravey, Dmitry, 2022. "Multilayer heat equations and their solutions via oscillating integral transforms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
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