  We establish the local asymptotic normality (LAN) property for estimating a
multidimensional parameter in the drift of a system of $N$ interacting
particles observed over a fixed time horizon in a mean-field regime $N
\rightarrow \infty$. By implementing the classical theory of Ibragimov and
Hasminski, we obtain in particular sharp results for the maximum likelihood
estimator that go beyond its simple asymptotic normality thanks to H\'ajek's
convolution theorem and strong controls of the likelihood process that yield
asymptotic minimax optimality (up to constants). Our structural results shed
some light to the accompanying nonlinear McKean-Vlasov experiment, and enable
us to derive simple and explicit criteria to obtain identifiability and
non-degeneracy of the Fisher information matrix. These conditions are also of
interest for other recent studies on the topic of parametric inference for
interacting diffusions.
