## 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 $$h$$.

### MSC:

 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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