论文标题
协方差运算符的平滑功能的估计：减少折刀偏见和有效等级的界限
Estimation of smooth functionals of covariance operators: jackknife bias reduction and bounds in terms of effective rank
论文作者
论文摘要
令$ e $为可分离的Banach空间，让$ x，x_1，\ dots，x_n，\ dots $ be i.i.d.高斯随机变量以$ e $为单位，平均零和未知的协方差操作员$σ：e^{\ ast} \ mapsto E. $基于观察值$ x_1，\ dots，\ dots，x_n $自然表征的$ q $σ$的估计复杂性是由$ $σ：$ c $ c = c的自然表征的。 \ frac {{{\ Mathbb e}_σ\ | x \ |^2} {\ | |σ\ |}，$，其中$ \ |σ\ | $是$σ的运算符标准。 $E^{\ast}$ into $E$ (equipped with the operator norm), our goal is to study the problem of estimation of $f(Σ)$ based on $X_1,\dots, X_n.$ The estimators of $f(Σ)$ based on jackknife type bias reduction are considered and the dependence of their Orlicz norm error rates on effective rank ${\bf r}(Σ),$ the sample size研究了$ n $和Hölder平滑度$ s $ f $的$ s $。特别是，如果$ {\ bf r}（σ）\ Lessim n^α$，对于某些$α\（0,1）$和$ s \ geq \ frac {1} {1-α}，则是$，则是古典$ \ sqrt {n} $ - 以及\ frac {1} {1-α}，$然后渐近正态性和所得估计器的渐近效率。以前，仅在有限维度欧几里得空间$ e = {\ Mathbb r}^d $的情况下获得此类型的结果（对于不同的估计器），而对于协方差操作员$σ$，其光谱从零界限为零（在这种情况下，$ {\ bf r}（\ bf r}（\ bf r}（x）（σ）\ asmp d $）。
Let $E$ be a separable Banach space and let $X, X_1,\dots, X_n, \dots$ be i.i.d. Gaussian random variables taking values in $E$ with mean zero and unknown covariance operator $Σ: E^{\ast}\mapsto E.$ The complexity of estimation of $Σ$ based on observations $X_1,\dots, X_n$ is naturally characterized by the so called effective rank of $Σ:$ ${\bf r}(Σ):= \frac{{\mathbb E}_Σ\|X\|^2}{\|Σ\|},$ where $\|Σ\|$ is the operator norm of $Σ.$ Given a smooth real valued functional $f$ defined on the space $L(E^{\ast},E)$ of symmetric linear operators from $E^{\ast}$ into $E$ (equipped with the operator norm), our goal is to study the problem of estimation of $f(Σ)$ based on $X_1,\dots, X_n.$ The estimators of $f(Σ)$ based on jackknife type bias reduction are considered and the dependence of their Orlicz norm error rates on effective rank ${\bf r}(Σ),$ the sample size $n$ and the degree of Hölder smoothness $s$ of functional $f$ are studied. In particular, it is shown that, if ${\bf r}(Σ)\lesssim n^α$ for some $α\in (0,1)$ and $s\geq \frac{1}{1-α},$ then the classical $\sqrt{n}$-rate is attainable and, if $s> \frac{1}{1-α},$ then asymptotic normality and asymptotic efficiency of the resulting estimators hold. Previously, the results of this type (for different estimators) were obtained only in the case of finite dimensional Euclidean space $E={\mathbb R}^d$ and for covariance operators $Σ$ whose spectrum is bounded away from zero (in which case, ${\bf r}(Σ)\asymp d$).
