arXiv 2108.002632021-08math.STstat.TH

对数凹位置族中的函数估计

Functional estimation in log-concave location families

Vladimir Koltchinskii, Martin Wahl

摘要：

设$\{P_θ:θ\in {\mathbb R}^d\}$是一个具有$P_θ(dx)=e^{-V(x-θ)}dx,$的对数凹位置族，其中$V:{\mathbb R}^d\mapsto {\mathbb R}$是一个已知的凸函数，$X_1,\dots, X_n$是i.i.d. r. v.从分布$P_θ$中采样，位置参数未知$θ.$目标是基于观测值$X_1,\dots, X_n.$估计光滑泛函$f:{\mathbb R}^d\mapsto {\mathbb R}$的值$f(θ)$在$V$足够光滑且$f$是来自Hölder空间$C^s,$中的球的泛函的情况下，我们开发具有由$L_2({\mathbb P}_θ)$测量的极小极大最优错误率的$f(θ)$的估计器。距离以及更一般的Orlicz范数距离。此外，我们证明了如果$d\leq n^α$和$s>\frac{1}{1-α},$，则所得估计在Hájek-LeCam意义下是渐近有效的，收敛速度为$\sqrt{n}.$.这推广了高斯移位模型中光滑泛函估计的早期结果.估计量具有$f_k(\hat θ),$的形式，其中$\hat θ$是最大似然估计量，而$f_k: {\mathbb R}^d\mapsto {\mathbb R}$（$k$取决于$s$）是根据$f$定义的泛函，旨在提供函数估计问题中的高阶偏差减少。偏差减少的方法是基于迭代参数自助法，它已成功地用于高斯模型的情况下之前。

英文摘要：

Let $\{P_θ:θ\in {\mathbb R}^d\}$ be a log-concave location family with $P_θ(dx)=e^{-V(x-θ)}dx,$ where $V:{\mathbb R}^d\mapsto {\mathbb R}$ is a known convex function and let $X_1,\dots, X_n$ be i.i.d. r.v. sampled from distribution $P_θ$ with an unknown location parameter $θ.$ The goal is to estimate the value $f(θ)$ of a smooth functional $f:{\mathbb R}^d\mapsto {\mathbb R}$ based on observations $X_1,\dots, X_n.$ In the case when $V$ is sufficiently smooth and $f$ is a functional from a ball in a Hölder space $C^s,$ we develop estimators of $f(θ)$ with minimax optimal error rates measured by the $L_2({\mathbb P}_θ)$-distance as well as by more general Orlicz norm distances. Moreover, we show that if $d\leq n^α$ and $s>\frac{1}{1-α},$ then the resulting estimators are asymptotically efficient in Hájek-LeCam sense with the convergence rate $\sqrt{n}.$ This generalizes earlier results on estimation of smooth functionals in Gaussian shift models. The estimators have the form $f_k(\hat θ),$ where $\hat θ$ is the maximum likelihood estimator and $f_k: {\mathbb R}^d\mapsto {\mathbb R}$ (with $k$ depending on $s$) are functionals defined in terms of $f$ and designed to provide a higher order bias reduction in functional estimation problem. The method of bias reduction is based on iterative parametric bootstrap and it has been successfully used before in the case of Gaussian models.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文