提出条件中心化统计量的渐近正态性，用于依赖随机场中的伪似然推断。
Pivotal CLTs for Pseudolikelihood via Conditional Centering in Dependent Random Fields
- 通过条件中心化和组合树剪枝法分析依赖场中统计量的分布。
- 在弱光滑条件下，统计量收敛到随机尺度的高斯混合，学生化后为枢轴量。
- 首次在密集非规则场景下获得伊辛模型参数联合中心极限定理，适用于复杂网络模型。
本文研究依赖随机场中形如 $N^{-1/2}igsum_{i=1}^N c_i(g(σ_i)-\mathbb{E}_N[g(σ_i)|σ_j,j\neq i])$ 的条件中心化统计量的波动行为，其中 $(σ_1,…,σ_N)$ 从依赖随机场中采样，$g$ 为有界函数。在对条件均值弱光滑性的假设下（涵盖稀疏与稠密相互作用），该统计量收敛于一个由二次方差和相互作用项决定的随机尺度高斯混合。经适当学生化后，极限为枢轴高斯分布。基于此理论，我们构建了依赖随机场中最大伪似然估计（MPLE）的通用渐近框架，并应用于具有成对及高阶相互作用的伊辛模型与指数随机图模型（ERGM）。特别地，首次在密集不规则区域获得逆温度与磁化强度参数的联合中心极限定理；并导出了无需限制在“亚临界”区域的条件边中心极限定理与边际MPLE中心极限定理。证明方法基于组合决策树剪枝的矩法，可能具独立研究价值。
原文摘要 · Abstract (English)
In this paper, we study fluctuations of conditionally centered statistics of the form $$N^{-1/2}\sum_{i=1}^N c_i(g(σ_i)-\mathbb{E}_N[g(σ_i)|σ_j,j\neq i])$$ where $(σ_1,\ldots ,σ_N)$ are sampled from a dependent random field, and $g$ is some bounded function. Our first main result shows that under weak smoothness assumptions on the conditional means (which cover both sparse and dense interactions), the above statistic converges to a Gaussian \emph{scale mixture} with a random scale determined by a \emph{quadratic variance} and an \emph{interaction component}. We also show that under appropriate studentization, the limit becomes a pivotal Gaussian. We leverage this theory to develop a general asymptotic framework for maximum pseudolikelihood (MPLE) inference in dependent random fields. We apply our results to Ising models with pairwise as well as higher-order interactions and exponential random graph models (ERGMs). In particular, we obtain a joint central limit theorem for the inverse temperature and magnetization parameters via the joint MPLE (to our knowledge, the first such result in dense, irregular regimes), and we derive conditionally centered edge CLTs and marginal MPLE CLTs for ERGMs without restricting to the ``sub-critical" region. Our proof is based on a method of moments approach via combinatorial decision-tree pruning, which may be of independent interest.
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