高维样本自协方差矩阵的尖峰特征值：CLT 及其应用

高维广义尖峰Fisher矩阵中尖峰数量检验的推断

Inference on testing the number of spikes in a high-dimensional generalized spiked Fisher matrix

本文针对放宽高斯假设与对角约束的高维广义尖峰Fisher矩阵，提出基于部分线性谱统计量的尖峰数量检验方法，建立中心极限定理，并应用于变量选择与变点检测，模拟与实证验证了方法的有效性和通用性。

具有发散尖峰的大维样本协方差矩阵线性谱统计量的中心极限定理

A CLT for the LSS of large dimensional sample covariance matrices with diverging spikes

本文将 Bai-Silverstein 定理推广到总体协方差矩阵含发散尖峰的情形，建立线性谱统计量的新中心极限定理，并应用于单位矩阵及广义尖峰模型检验，证明其渐近功效通常高于 Roy 最大根检验。

椭圆分布下样本协方差矩阵预解式双线性形式的极限行为及其应用

Limiting behavior of bilinear forms for the resolvent of sample covariance matrices under elliptical distribution with applications

本文为椭圆分布下样本协方差矩阵预解式的双线性形式建立联合中心极限定理，发现极限参数的相变，并应用于特征向量统计和尖峰特征值，推广了Haar猜想。

CLT for LSS of sample covariance matrices with unbounded dispersions

Under the high-dimensional setting that data dimension and sample size tend to infinity proportionally, we derive the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix. Different from existing literature, our results do not require the assumption that the population covariance matrices are bounded. Moreover, many common kernel functions in the real data such as logarithmic functions and polynomial functions are allowed in this paper. In our model, the number of spiked eigenvalues can be fixed or tend to infinity. One salient feature …

广义尖峰模型下四种线性假设检验统计量的渐近分布

Asymptotic distributions of four linear hypotheses test statistics under generalized spiked model

该研究建立大维广义尖峰样本协方差矩阵线性谱统计量的中心极限定理，推导四种线性假设检验统计量的渐近分布，用泰勒展开逼近围道积分，给出三种检验的渐近功效并与Roy最大根检验对比，经数值模拟验证了逼近准确性。

On the sphericity test with large-dimensional observations

In this paper, we propose corrections to the likelihood ratio test and John's test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with…

Spiked eigenvalues of noncentral Fisher matrix with applications

In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$, where ${\mathbf C}_n$ is a noncentral sample covariance matrix defined by $(\mathbf Ξ+\mathbf X)(\mathbf Ξ+\mathbf X)^*/n$ and $\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$. The matrices $\mathbf X$ and $\mathbf Y$ are two independent {Gaussian} arrays, with respective $p\times n$ and $p\times N$ and the Gaussian entries of them are \textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $…

On the Eigenstructure of Covariance Matrices with Divergent Spikes

For a generalization of Johnstone's spiked model, a covariance matrix with eigenvalues all one but $M$ of them, the number of features $N$ comparable to the number of samples $n: N=N(n), M=M(n), γ^{-1} \leq \frac{N}{n} \leq γ$ where $γ\in (0,\infty),$ we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever $M$ grows slightly slower than $n: \lim_{n \to \infty}{\frac{\sqrt{\log{n}}}{\log{\frac{n}{M(n)}}}}=0.$ Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1/6})$ an…

Sharp detection in PCA under correlations: all eigenvalues matter

Principal component analysis (PCA) is a widely used method for dimension reduction. In high dimensional data, the "signal" eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the "noise" eigenvalues. Therefore, popular tests based on the largest eigenvalue have little power to detect weak PCs. In the special case of the spiked model, certain tests asymptotically equivalent to linear spectral statistics (LSS)---averaging effects over all eigenvalues---were recently shown to achieve some power. We consider a nonparametric, non-Gaussian genera…