We prove finite-sample concentration and anti-concentration bounds for dimension estimation using Gaussian kernel sums. Our bounds provide explicit dependence on sample size, bandwidth, and local geometric and distributional parameters, characterizing precisely how regularity conditions influence statistical performance. We also propose a bandwidth selection heuristic using derivative information, supported by numerical experiments.

翻译：我们证明了使用高斯核和进行维度估计的有限样本集中与反集中界。我们的界明确依赖于样本大小、带宽以及局部几何与分布参数，精确刻画了正则性条件如何影响统计性能。我们还提出了一种利用导数信息的带宽选择启发式方法，并通过数值实验予以验证。