标题： 一个简单自适应的密度平方积分估计器 显示英文标题

标题： A simple adaptive estimator of the integrated square of a density

摘要： 给定一个独立同分布样本$X_1,...,X_n$，其具有属于实数线上阶数为$\alpha$的Sobolev空间的公共有界密度$f_0$，考虑二次泛函$\int_{\mathbb{R}}f_0^2(x) \mathrm{d}x$的估计问题。结果显示，如果$\alpha>1/4$，则基于核的最简单插件估计量\[\frac{2}{n(n-1)h_n}\sum_{1\leq i<j\leq n}K\biggl(\frac{X_i-X_j}{h_n}\biggr)\]是渐近有效的；如果$\alpha\le1/4$，则它是速率最优的。 随后提出了一种基于数据驱动的规则来选择带宽$h_n$，它不依赖于$\alpha$的先验知识，从而使得相应的估计量对于$\alpha \leq1/4$是速率自适应的，并且在$\alpha>1/4$成立时渐近有效。 显示英文摘要

摘要： Given an i.i.d. sample $X_1,...,X_n$ with common bounded density $f_0$ belonging to a Sobolev space of order $\alpha$ over the real line, estimation of the quadratic functional $\int_{\mathbb{R}}f_0^2(x) \mathrm{d}x$ is considered. It is shown that the simplest kernel-based plug-in estimator \[\frac{2}{n(n-1)h_n}\sum_{1\leq i<j\leq n}K\biggl(\frac{X_i-X_j}{h_n}\biggr)\] is asymptotically efficient if $\alpha>1/4$ and rate-optimal if $\alpha\le1/4$. A data-driven rule to choose the bandwidth $h_n$ is then proposed, which does not depend on prior knowledge of $\alpha$, so that the corresponding estimator is rate-adaptive for $\alpha \leq1/4$ and asymptotically efficient if $\alpha>1/4$.