This paper examines the limiting variance of nearest neighbor matching estimators for average treatment effects with a fixed number of matches. We present, for the first time, a closed-form expression for this limit. Here the key is the establishment of the limiting second moment of the catchment area's volume, which resolves a question of Abadie and Imbens. At the core of our approach is a new universality theorem on the measures of high-order Voronoi cells, extending a result by Devroye, Gy\"orfi, Lugosi, and Walk.

翻译：本文考察了固定匹配数下最近邻匹配估计量对于平均处理效应的极限方差。我们首次给出了该极限的闭式表达式。关键在于建立了捕获区域体积极限二阶矩，这解决了Abadie和Imbens提出的一个问题。我们方法的核心是关于高阶Voronoi单元测度的新普适性定理，该定理扩展了Devroye、Györfi、Lugosi和Walk的结果。