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 $1$.

翻译：我们全面描述了偏斜布朗运动（SBM）偏度参数的最大似然估计量（MLE）的渐近性质。基于近期关于SBM估计量收敛速率的中心极限定理结果，我们证明了一个未解猜想：MLE渐近地服从涉及局部时间的混合正态分布，其收敛速率达到$1/4$阶。我们还给出了MLE的级数展开，并研究了得分函数及其导数的渐近行为，以及它们随偏度参数的变化规律。特别地，我们揭示了当SBM实际为布朗运动时的特殊行为，并量化了偏度参数接近$-1$或$1$时展开系数的爆炸性增长。