In the regression framework, the empirical measure based on the responses resulting from the nearest neighbors, among the covariates, to a given point $x$ is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a local bracketing entropy condition on the underlying class of functions reflecting the localizing nature of the nearest neighbor algorithm. Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is simply equal to the conditional covariance operator given the covariate value. This suggests the possibility of using standard formulas to estimate the variance by using only the nearest neighbors instead of the full data. This is illustrated on two problems: the estimation of the conditional cumulative distribution function and local linear regression.

翻译：在回归框架下，基于协变量中给定点 $x$ 的最近邻响应所构建的经验测度被引入并作为核心统计量进行研究。首先，在反映最近邻算法局部特性的函数类上，若满足局部括号熵条件，则相关经验过程满足一致中心极限定理。其次，在均匀熵数的经典条件下（通常称为Vapnik-Chervonenkis条件），建立了均匀非渐近界。一致中心极限定理中高斯极限的协方差恰好等于给定协变量值条件下的条件协方差算子。这表明可以利用标准公式仅通过最近邻而非全部数据来估计方差。该结论通过两个问题得到验证：条件累积分布函数的估计与局部线性回归。