In this paper we consider the problem of parameter estimation in the $p$-spin Curie-Weiss model, for $p \geq 3$. We provide a complete description of the limiting properties of the maximum likelihood (ML) estimates of the inverse temperature and the magnetic field given a single realization from the $p$-spin Curie-Weiss model, complementing the well-known results in the 2-spin case (Comets and Gidas (1991)). Our results unearth various new phase transitions and surprising limit theorems, such as the existence of a 'critical' curve in the parameter space, where the limiting distribution of the ML estimates is a mixture with both continuous and discrete components. The number of mixture components is either two or three, depending on, among other things, the sign of one of the parameters and the parity of $p$. Another interesting revelation is the existence of certain 'special' points in the parameter space where the ML estimates exhibit a superefficiency phenomenon, converging to a non-Gaussian limiting distribution at rate $N^{\frac{3}{4}}$. Using these results we can obtain asymptotically valid confidence intervals for the inverse temperature and the magnetic field at all points in the parameter space where consistent estimation is possible.

翻译：在本文中,我们考虑了美元螺旋螺旋螺旋螺旋螺旋螺旋螺旋体模型的参数估计问题,为美元=3美元。我们完整地描述了对反温和磁场最大可能性(ML)估计值的最大值(ML)与反温和离散成分的混合值的有限性。根据美元螺旋螺旋螺旋体模型的一次性实现,混合成分的数量为两个或三个,这主要取决于参数之一的标志和美元对等值。另一个有趣的发现是参数空间存在某些“特殊”点,在参数空间中,ML估计显示一种超效现象,并令人惊讶地限制理论,将参数空间中“临界”曲线的分布与参数空间中的“临界”曲线相融合,在参数空间中限制以美元=%xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx