Asymptotics of Maximum Likelihood Estimators for the Curie-Weiss Model

Abstract

We study the asymptotics of the ML estimators for the Curie-Weiss model parametrized by the inverse temperature $\beta$ and the external field $h$. We show that if both $\beta$ and $h$ are unknown, the ML estimator of $(\beta, h)$ does not exist. For $\beta$ known, the ML estimator $\hat{h}_n$ of $h$ exhibits, at a first order phase transition point, superefficiency in the sense that its asymptotic variance is half of that of nearby points. At the critical point $(\beta = 1)$, if the true value is $h = 0$, then $n^{3/4}\hat h_n$ has a non-Gaussian limiting law. Away from phase transition points, $\hat h_n$ is asymptotically normal and efficient. We also study the asymptotics of the ML estimator of $\beta$ for known $h$.