论文标题
密度集合的经验估计器的立方根弱收敛性
Cube root weak convergence of empirical estimators of a density level set
论文作者
论文摘要
给定$ n $独立的随机向量，具有$ \ mathbb {r}^d $上的常见密度$ f $，我们研究了三个基于经验测量的基于经验测量的估计量的弱收敛性$L_λ$。由于这些设置值估计器接近$l_λ$，因此即使是其弱收敛的表述也不标准。我们以$ n^{ - 1/3} $的价格确定$L_λ$的对称差异的联合限制分布。事实证明，最小体积集和最大概率集估计器在渐近上是无法区分的，而多余的质量集估计器表现出“更丰富”的极限行为。参数依赖于边界本地经验过程，其气缸表示，$l_λ$边界周围的无维度浓度以及漂移的维也纳过程的设定值。
Given $n$ independent random vectors with common density $f$ on $\mathbb{R}^d$, we study the weak convergence of three empirical-measure based estimators of the convex $λ$-level set $L_λ$ of $f$, namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets $\mathcal{A}$ that contains $L_λ$. Since these set-valued estimators approach $L_λ$, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of $L_λ$ and each of the three estimators, at rate $n^{-1/3}$. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits "richer" limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of $L_λ$, and the set-valued argmax of a drifted Wiener process.
