  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 $1$.
