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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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