论文标题
高维模型中平滑功能的估计：自举链和高斯近似
Estimation of smooth functionals in high-dimensional models: bootstrap chains and Gaussian approximation
论文作者
论文摘要
令$ x^{（n）} $为从分布中取样的观察$p_θ^{（n）} $，带有未知参数$θ，$ $ $θ$是Banach Space $ e $中的向量（最经常是尺寸的高维空间$ d $）。我们研究功能性$ f的$ f（θ）$的估计问题：e \ mapsto {\ mathbb r} $，基于观察$ x^{（n）} \ simp_θ^{（n）}的某些平滑度$ s> 0 $ $ s> 0 $参数$θ$使得$ \ sqrt {n}（\hatθ_n-θ）$在分布中足够接近到$ e的平均零高斯随机向量，$我们构建功能性$ g：e \ mapSto {\ mathbb r} $ $ \ sqrt {n} $ rates提供$ s> \ frac {1} {1-α} $和$ d \ d \ d \ leq n^α$的某些$α\ in（0,1）。$我们还得出了Orlicz norm误差的一般上限，用于估计器$ g（\ hat $ diff sploomisensive $ dimestive $ diff SpliseSive $ diff SpliseSival $ dife y dimife $ dife y dimefience $ dife n dimife $ n j $ sploomise $ n j $ n j $ s $ n n o' $ \ sqrt {n}的近似（\ hatθ_n-θ）。尤其是，在某些高维指数模型中，此方法会产生渐近有效的估计器。
Let $X^{(n)}$ be an observation sampled from a distribution $P_θ^{(n)}$ with an unknown parameter $θ,$ $θ$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(θ)$ for a functional $f:E\mapsto {\mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}\sim P_θ^{(n)}.$ Assuming that there exists an estimator $\hat θ_n=\hat θ_n(X^{(n)})$ of parameter $θ$ such that $\sqrt{n}(\hat θ_n-θ)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:E\mapsto {\mathbb R}$ such that $g(\hat θ_n)$ is an asymptotically normal estimator of $f(θ)$ with $\sqrt{n}$ rate provided that $s>\frac{1}{1-α}$ and $d\leq n^α$ for some $α\in (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(\hat θ)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $\sqrt{n}(\hat θ_n-θ).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.
