发表机构
Philipps-Universität Marburg; Universität Rostock（马尔堡菲利普斯大学; 罗斯托克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种局部多项式密度比估计器，在仅假设比值光滑时达到极小极大最优速率，支持边界点，并提供浓度不等式、导数估计、渐近正态性及KL信息估计。
AI 中文摘要
我们提出了一种新颖的局部多项式估计器，用于估计两个 d 维密度 f 和 g 的比值 r=f/g，其中 f 和 g 的独立样本是可用的。该估计器被证明在任意光滑度指数的 Hölder 类上达到逐点极小极大最优速率，且无需额外的对数因子，并且仅假设 r 的光滑性，而不假设 f 或 g 的光滑性。在关于（未知）边界的温和几何假设下，我们的分析对于与 g 相关的分布支撑边界上的点仍然有效。我们还推导了该估计器的浓度不等式，这在分类应用中可能有用，并给出了上确界范数下的速率，其中上确界也取遍边界支撑点。获得该速率的光滑类足够大，以至于个体密度 f 和 g 不能在此类上一致地估计。我们还提供了 r 的偏导数的直接估计器及其收敛速率。我们在相当一般的假设下获得了估计器的渐近正态性，并且同样包括边界点，同时提供了一致方差估计器，允许数据驱动的学生化。此外，我们展示了如何使用我们的估计器来估计 Kullback-Leibler 信息，在构造中额外使用了去偏技术。在 r 相对于维度 d 足够光滑的条件下，我们提供了参数速率和渐近正态性。
英文摘要
We propose a novel local-polynomial estimator of the ratio $r=f/g$ of two $d$-dimensional densities $f$ and $g$, from which independent samples are available. The estimator is shown to achieve pointwise minimax optimal rates over Hölder classes of arbitrary smoothness index without additional logarithmic factors, and with smoothness being only assumed of $r$ but not of $f$ nor $g$. Our analysis remains valid for points on the boundary of the support of the distribution associated to $g$ under a mild geometric assumption on the (unknown) boundary. We also derive a concentration inequality for the estimator, which can be useful in applications to classification, and give a rate in the supremum norm, where the supremum is also taken over boundary support points. The smoothness class over which the rate is obtained is sufficiently large that the individual densities $f$ and $g$ cannot be consistently estimated uniformly over this class. Direct estimators of the partial derivatives of $r$ together with rates of convergence are also provided. We also obtain asymptotic normality of the estimators under quite general assumptions and again including boundary points, with a consistent variance estimator allowing for data-driven studentization. Moreover, we show how our estimator can be used to estimate the Kullback-Leibler information, in a construction which additionally uses debiasing. We provide the parametric rate together with asymptotic normality under sufficient smoothness of $r$ relative to the dimension $d$.
