Maximum likelihood estimator for skew Brownian motion: the convergence rate

Antoine Lejay; Sara Mazzonetto

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arXiv · 2302.02954

Maximum likelihood estimator for skew Brownian motion: the convergence rate

Antoine Lejay·Sara Mazzonetto

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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 $1$.

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Antoine Lejay, Sara Mazzonetto. 2023-02-06. Maximum likelihood estimator for skew Brownian motion: the convergence rate. https://arxiv.org/abs/2302.02954

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