  We give an asymptotic development of the maximum likelihood estimator (MLE),
or any other estimator defined implicitly, in a way which involves the limiting
behavior of the score and its higher-order derivatives. This development, which
is explicitly computable, gives some insights about the non-asymptotic behavior
of the renormalized MLE and its departure from its limit. We highlight that the
results hold whenever the score and its derivative converge, including to non
Gaussian limits. Our approach is based on an asymptotic implicit function
theorem, inspired from perturbative approaches.
