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，本文引入并研究了基于协变量中某给定点$x$的最近邻所对应的响应变量的经验测度，将其作为核心统计量。首先，在反映最近邻算法局部化特性的底层函数类的局部括号熵条件下，证明了相关的经验过程满足一致中心极限定理。其次，在关于均匀熵数的一个著名条件（通常称为Vapnik-Chervonenkis条件）下，建立了统一的非渐近界。一致中心极限定理中得到的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。这表明可以利用仅基于最近邻而非全部数据的标准公式来估计方差。这一方法在估计条件累积分布函数和局部线性回归两个问题中得到了验证。