In this paper, we derive results about the limiting distribution of the empirical magnetization vector and the maximum likelihood (ML) estimates of the natural parameters in the tensor Curie-Weiss Potts model. Our results reveal surprisingly new phase transition phenomena including the existence of a smooth curve in the interior of the parameter plane on which the magnetization vector and the ML estimates have mixture limiting distributions, the latter comprising of both continuous and discrete components, and a surprising superefficiency phenomenon of the ML estimates, which stipulates an $N^{-3/4}$ rate of convergence of the estimates to some non-Gaussian distribution at certain special points of one type and an $N^{-5/6}$ rate of convergence to some other non-Gaussian distribution at another special point of a different type. The last case can arise only for one particular value of the tuple of the tensor interaction order and the number of colors. These results are then used to derive asymptotic confidence intervals for the natural parameters at all points where consistent estimation is possible.

翻译：本文研究了张量Curie-Weiss Potts模型中经验磁化向量的极限分布以及自然参数的最大似然估计。我们的结果揭示了令人惊讶的新相变现象，包括在参数平面内部存在一条光滑曲线，在该曲线上磁化向量和最大似然估计具有混合极限分布（包含连续分量和离散分量），以及一个令人意外的最大似然估计超效率现象：在某些特殊类型点上，估计量以$N^{-3/4}$的收敛速度趋于非高斯分布；在另一不同类型特殊点上，则以$N^{-5/6}$的收敛速度趋于另一种非高斯分布。最后一种情况仅当张量相互作用阶数与颜色数的元组取特定值时才会出现。基于这些结果，我们在所有可能实现一致估计的点上推导了自然参数的渐近置信区间。