ASYMPTOTICS OF MAXIMUM-LIKELIHOOD ESTIMATORS FOR THE CURIE-WEISS MODEL

We study the asymptotics of the ML estimators for the Curie-Weiss model parameterized 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 h triple-over-dot 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 n3/4h triple-over-dot n has a non-Gaussian limiting law. Away from phase transition points, h triple-over-dot n is asymptotically normal and efficient. We also study the asymptotics of the ML estimator of beta-for known h.