Neighborhood smoothing methods achieve minimax-optimal rates for estimating edge probabilities under graphon models, but their use for statistical inference has remained limited. The main obstacle is that classical neighborhood smoothers select data-driven neighborhoods and average edges using the same adjacency matrix, inducing complex dependencies that invalidate standard concentration and normal approximation arguments. We introduce a leave-one-out modification of neighborhood smoothing for undirected simple graphs. When estimating a single entry P_ij, the neighborhood of node i is constructed from an adjacency matrix in which the jth row and column are set to zero, thereby decoupling neighborhood selection from the edges being averaged. We show that this construction restores conditional independence of the centered summands, enabling the use of classical probabilistic tools for inference. Under piecewise Lipschitz graphon assumptions and logarithmic degree growth, we derive variance-adaptive concentration inequalities based on Bousquet's inequality and establish Berry-Esseen bounds with explicit rates for the normalized estimation error. These results yield both finite-sample and asymptotic confidence intervals for individual edge probabilities. The same leave-one-out structure also supports an honest cross-validation scheme for tuning parameter selection, for which we prove an oracle inequality. The proposed estimator retains the optimal row-wise mean-squared error rates of classical neighborhood smoothing while providing valid entrywise uncertainty quantification.

翻译：邻域平滑方法在图函数模型下估计边概率时达到了极小极大最优速率，但其在统计推断中的应用仍较为有限。主要障碍在于经典邻域平滑器使用同一邻接矩阵选择数据驱动的邻域并对边进行平均，从而引入了复杂的依赖关系，使得标准集中性与正态逼近论证失效。本文针对无向简单图提出一种留一法改进的邻域平滑方法。当估计单个条目P_ij时，节点i的邻域通过将邻接矩阵第j行和第j列置零来构建，从而将邻域选择与待平均的边解耦。我们证明这种构造恢复了中心化加项的条件独立性，使得经典概率推断工具得以应用。在分段Lipschitz图函数假设与对数阶度增长的条件下，我们基于Bousquet不等式推导了方差自适应的集中不等式，并为归一化估计误差建立了具有显式速率的Berry-Esseen界。这些结果同时给出了针对单一边概率的有限样本与渐近置信区间。相同的留一法结构还支持用于调参选择的诚实交叉验证方案，并为此证明了oracle不等式。所提估计器在保持经典邻域平滑最优行向均方误差速率的同时，提供了有效的逐条目不确定性量化。