样本协方差谱投影子的正态逼近与集中
统计理论
2015-04-29 v1 统计理论
摘要
设 X , X 1 , … , X n X,X_1,\dots, X_n X , X 1 , … , X n H {\mathbb H} H Σ = E ( X ⊗ X ) , \Sigma={\mathbb E}(X\otimes X), Σ = E ( X ⊗ X ) , Σ ^ : = n − 1 ∑ j = 1 n ( X j ⊗ X j ) \hat \Sigma:=n^{-1}\sum_{j=1}^n (X_j\otimes X_j) Σ ^ := n − 1 ∑ j = 1 n ( X j ⊗ X j ) ( X 1 , … , X n ) (X_1,\dots, X_n) ( X 1 , … , X n ) P r P_r P r Σ \Sigma Σ r r r μ r \mu_r μ r P ^ r \hat P_r P ^ r P r P_r P r sup x ∈ R ∣ P { ∥ P ^ r − P r ∥ 2 2 − E ∥ P ^ r − P r ∥ 2 2 V a r 1 / 2 ( ∥ P ^ r − P r ∥ 2 2 ) ≤ x } − Φ ( x ) ∣ \sup_{x\in {\mathbb R}} \left|{\mathbb P}\left\{\frac{\|\hat P_r-P_r\|_2^2-{\mathbb E}\|\hat P_r-P_r\|_2^2}{{\rm Var}^{1/2}(\|\hat P_r-P_r\|_2^2)}\leq x\right\}-\Phi(x)\right| x ∈ R sup P { Var 1/2 ( ∥ P ^ r − P r ∥ 2 2 ) ∥ P ^ r − P r ∥ 2 2 − E ∥ P ^ r − P r ∥ 2 2 ≤ x } − Φ ( x ) ∥ ⋅ ∥ 2 \|\cdot\|_2 ∥ ⋅ ∥ 2 Φ \Phi Φ Σ \Sigma Σ r ( Σ ) = t r ( Σ ) ∥ Σ ∥ ∞ , {\bf r}(\Sigma)=\frac{{\rm tr}(\Sigma)}{\|\Sigma\|_{\infty}}, r ( Σ ) = ∥Σ ∥ ∞ tr ( Σ ) , t r ( Σ ) {\rm tr}(\Sigma) tr ( Σ ) Σ \Sigma Σ ∥ Σ ∥ ∞ \|\Sigma\|_{\infty} ∥Σ ∥ ∞ V a r ( ∥ P ^ r − P r ∥ 2 2 ) {\rm Var}(\|\hat P_r-P_r\|_2^2) Var ( ∥ P ^ r − P r ∥ 2 2 ) E ∥ P ^ r − P r ∥ 2 2 {\mathbb E}\|\hat P_r-P_r\|_2^2 E ∥ P ^ r − P r ∥ 2 2 V a r ( ∥ P ^ r − P r ∥ 2 2 ) {\rm Var}(\|\hat P_r-P_r\|_2^2) Var ( ∥ P ^ r − P r ∥ 2 2 ) ∥ P ^ r − P r ∥ 2 2 \|\hat P_r-P_r\|_2^2 ∥ P ^ r − P r ∥ 2 2
引用
@article{arxiv.1504.07333,
title = {Normal approximation and concentration of spectral projectors of sample covariance},
author = {Vladimir Koltchinskii and Karim Lounici},
journal= {arXiv preprint arXiv:1504.07333},
year = {2015}
}
备注
arXiv admin note: text overlap with arXiv:1408.4643