AI 中文总结
本文针对高维立方张量吉布斯测度的联合参数估计问题，以最大伪似然估计器为对象，推导了相关模型的一致性与病态性条件，发展了非线性大偏差等工具，为ERGMs等模型的参数估计提供了理论支撑。
AI 中文摘要
本文受密集ERGMs、算术级数模型和非均匀随机超图的启发，研究了具有立方张量相互作用的高维吉布斯测度中基于单次观测的联合参数估计问题。针对最大伪似然估计器，本文给出了联合一致性和渐近病态性的可验证条件：对于边-三角形ERGM，在非负场的铁磁区域伪似然病态，但在足够强的反铁磁区域一致；对于边-三星ERGM，其在所有逆温度和外场下均病态。本文还研究了算术级数模型和非均匀超图模型的一致性问题，证明过程发展了适用于立方张量吉布斯测度的非线性大偏差和平均场近似工具，具有广泛应用前景。
英文摘要
We study joint parameter estimation from a single observation in high-dimensional Gibbs measures with cubic tensor interactions, motivated by dense ERGMs, arithmetic-progression models, and inhomogeneous random hypergraphs. Focusing on the maximum pseudolikelihood estimator, we give checkable conditions for joint consistency and asymptotic ill-conditioning. For the edge-triangle ERGM, pseudolikelihood is ill-conditioned in the ferromagnetic regime with nonnegative field, but consistent in a sufficiently strong antiferromagnetic regime. For the edge-three-star ERGM, it is ill-conditioned for all inverse temperatures and external fields. We also study consistency for arithmetic-progression, and inhomogeneous hypergraph models. Our proofs develop nonlinear large-deviation and mean-field approximation tools for cubic tensor Gibbs measures, which have scope for broad applications.
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