Asymptotics of maximum likelihood estimators for the Curie-Weiss model. (English) Zbl 0749.62018

Summary: 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 \(h\).


62F12 Asymptotic properties of parametric estimators
62P99 Applications of statistics
62M40 Random fields; image analysis
82B99 Equilibrium statistical mechanics
82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium)
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