We discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and,more generally, $M$-estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set $S$.We give a counterexample showing that regularity of $S$ in the sense of Clarke is not sufficient for asymptotic equivalence of $\sqrt{n}$-consistent locally optimal $M$-estimators.We argue further that stronger properties, such as so-called near convexity or prox-regularity of $S$ are required in order to ensure that any two $\sqrt{n}$-consistent locally optimal $M$-estimators have the same asymptotics.
