论文标题
大样本精度矩阵的对角线条目的波动
Fluctuations of the diagonal entries of a large sample precision matrix
论文作者
论文摘要
对于给定的$ p \ times n $ data矩阵$ \ textbf {x} _n $ with I.I.D。中心条目和人口协方差矩阵$ \bfς$，相应的样本精度矩阵$ \ hat {\bfς}^{ - 1} $定义为样本协方差矩阵$ \ hat fat fhat {\ hat \ textbf {x} _n \ textbf {x} _n^\ top \bfς^{1/2} $。我们在这种情况下确定了矩阵$ \ hat {\bfς}^{ - 1} $的对角线条目向量的联合分布，其中$ p_n = p <n $和$ p/n \ in [0,1）$ in [0,1）$ for $ n \ for $ n \ for $ n \ to for for for for $ n \ to for for for for for for for for for for for for for for $ n \ to ftto ftty $ n \ for ftty uftty $和$ \ bfdy $ \bfπ$ a diagonal matrix。值得注意的是，我们的结果涵盖了尺寸与样本量相比及其相同大小相比的情况。我们的方法基于数据矩阵的QR分解，得出与随机二次形式的连接，并允许将中心限制定理应用于martingale差异方案。此外，我们讨论了与样品协方差矩阵的线性光谱统计的有趣联系。更确切地说，样本精度矩阵的对数对数进入可以解释为$ \ hat {\bfς} $的两个高度依赖的线性光谱统计量和$ \ hat {\ hat {\bfςd的subsatrix的差异。光谱统计的这种差异比每个单个统计量都比规模波动得多。
For a given $p\times n$ data matrix $\textbf{X}_n$ with i.i.d. centered entries and a population covariance matrix $\bfΣ$, the corresponding sample precision matrix $\hat{\bfΣ}^{-1}$ is defined as the inverse of the sample covariance matrix $\hat{\bfΣ} = (1/n) \bfΣ^{1/2} \textbf{X}_n\textbf{X}_n^\top \bfΣ^{1/2}$. We determine the joint distribution of a vector of diagonal entries of the matrix $\hat{\bfΣ}^{-1}$ in the situation, where $p_n=p< n$ and $p/n \to y \in [0,1)$ for $n\to\infty$ and $\bfΣ$ is a diagonal matrix. Remarkably, our results cover both the case where the dimension is negligible in comparison to the sample size and the case where it is of the same magnitude. Our approach is based on a QR-decomposition of the data matrix, yielding a connection to random quadratic forms and allowing the application of a central limit theorem for martingale difference schemes. Moreover, we discuss an interesting connection to linear spectral statistics of the sample covariance matrix. More precisely, the logarithmic diagonal entry of the sample precision matrix can be interpreted as a difference of two highly dependent linear spectral statistics of $\hat{\bfΣ}$ and a submatrix of $\hat{\bfΣ}$. This difference of spectral statistics fluctuates on a much smaller scale than each single statistic.
