  The Maximum Likelihood Estimator (MLE) serves an important role in statistics
and machine learning. In this article, for i.i.d. variables, we obtain
constant-specified and sharp concentration inequalities and oracle inequalities
for the MLE only under exponential moment conditions. Furthermore, in a robust
setting, the sub-Gaussian type oracle inequalities of the log-truncated maximum
likelihood estimator are derived under the second-moment condition.
