论文标题
中央限制定理和近乎经典的浆果 - 自我标准化总和高尺寸
Central Limit Theorem and Near classical Berry-Esseen rate for self normalized sums in high dimensions
论文作者
论文摘要
在本文中，我们对$ t_n = \ big（\ sum_ {i = 1}^{n} x_ {i1}/\ big（\ sqrt {\ sum_ {\ sum_ {i = 1） $ \ sum_ {i = 1}^{n} x_ {ip}/\ big（\ sqrt {\ sum_ {\ sum_ {i = 1}^{n}^{n} x_ {ip}^2}^2}^2} \ big）\ big） $ \ Mathcal {a}^{re} = \ {\ prod_ {j = 1}^{p} [a_j，b_j，b_j] \ cap \ mathcal {r}： - \ \ \ iffty \ leq a_j \ leq a_j \ leq b_j \ leq b_j \ leq \ leq \ leq \ leq \ leq \ eftty，j $ x_1，\ dots，x_n $是非脱位独立的$ p- $尺寸随机向量。我们假设$ x_i $的组件是独立的且分布式的（IID），并研究了$ \ t_n $ to $ \ t_n $ over $ \ mathcal {a} a}^{re} $在统一中央限制定理（UCLT）中的最佳截止率。目的是减少指数矩条件，通常假定将尺寸相对于样本量在高维CLT中的指数生长到某些多项式力矩条件。确实，我们确定只有[2,4] $中的某种多项式订单$ \的存在才足以达到$ p $的指数增长。但是，即使$ o \ big（n^{1/2} \ big）$的增长率无法进一步提高$ n $的功率，即使$ x_ {ij {ij} $是跨$（i，j）$和$ x_ {11} $的IID的功率。我们还建立了$ -n^{ - κ/2} $ berry-esseen的$ t_n $的$ t_n $，在$（2+κ）的存在下，$ x_ {ij} $的绝对矩$ 0 <κ\ be \ leq 1 $。当$κ= 1 $时，获得的浆果率也被证明是最佳的。作为一个应用程序，我们找到了组件的T统计量的各个版本，这在高维统计推断中可能很有用。
In this article, we are interested in the high dimensional normal approximation of $T_n =\Big(\sum_{i=1}^{n}X_{i1}/\Big(\sqrt{\sum_{i=1}^{n}X_{i1}^2}\Big),\dots,$ $\sum_{i=1}^{n}X_{ip}/\Big(\sqrt{\sum_{i=1}^{n}X_{ip}^2}\Big)\Big)$ in $\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leq b_j \leq \infty, j=1,\ldots,p\}$, where $X_1,\dots,X_n$ are non-degenerate independent $p-$dimensional random vectors. We assume that the components of $X_i$ are independent and identically distributed (iid) and investigate the optimal cut-off rate of $\log p$ in the uniform central limit theorem (UCLT) for $T_n$ over $\mathcal{A}^{re}$. The aim is to reduce the exponential moment conditions, generally assumed for exponential growth of the dimension with respect to the sample size in high dimensional CLT, to some polynomial moment conditions. Indeed, we establish that only the existence of some polynomial moment of order $\in [2,4]$ is sufficient for exponential growth of $p$. However the rate of growth of $\log p$ can not further be improved from $o\big(n^{1/2}\big)$ as a power of $n$ even if $X_{ij}$'s are iid across $(i,j)$ and $X_{11}$ is bounded. We also establish near$-n^{-κ/2}$ Berry-Esseen rate for $T_n$ in high dimension under the existence of $(2+κ)$th absolute moments of $X_{ij}$ for $0< κ\leq 1$. When $κ=1$, the obtained Berry-Esseen rate is also shown to be optimal. As an application, we find respective versions for component-wise Student's t-statistic, which may be useful in high dimensional statistical inference.
