We consider the extreme eigenvalues of the sample covariance matrix $Q=YY^*$ under the generalized elliptical model that $Y=\Sigma^{1/2}XD.$ Here $\Sigma$ is a bounded $p \times p$ positive definite deterministic matrix representing the population covariance structure, $X$ is a $p \times n$ random matrix containing either independent columns sampled from the unit sphere in $\mathbb{R}^p$ or i.i.d. centered entries with variance $n^{-1},$ and $D$ is a diagonal random matrix containing i.i.d. entries and independent of $X.$ Such a model finds important applications in statistics and machine learning. In this paper, assuming that $p$ and $n$ are comparably large, we prove that the extreme edge eigenvalues of $Q$ can have several types of distributions depending on $\Sigma$ and $D$ asymptotically. These distributions include: Gumbel, Fr\'echet, Weibull, Tracy-Widom, Gaussian and their mixtures. On the one hand, when the random variables in $D$ have unbounded support, the edge eigenvalues of $Q$ can have either Gumbel or Fr\'echet distribution depending on the tail decay property of $D.$ On the other hand, when the random variables in $D$ have bounded support, under some mild regularity assumptions on $\Sigma,$ the edge eigenvalues of $Q$ can exhibit Weibull, Tracy-Widom, Gaussian or their mixtures. Based on our theoretical results, we consider two important applications. First, we propose some statistics and procedure to detect and estimate the possible spikes for elliptically distributed data. Second, in the context of a factor model, by using the multiplier bootstrap procedure via selecting the weights in $D,$ we propose a new algorithm to infer and estimate the number of factors in the factor model. Numerical simulations also confirm the accuracy and powerfulness of our proposed methods and illustrate better performance compared to some existing methods in the literature.

翻译：考虑广义椭圆模型下样本协方差矩阵 $Q=YY^*$ 的极端特征值问题，其中 $Y=\Sigma^{1/2}XD.$ 这里 $\Sigma$ 为有界 $p \times p$ 正定确定性矩阵，代表总体协方差结构；$X$ 是 $p \times n$ 随机矩阵，其独立列向量取自 $\mathbb{R}^p$ 单位球面或具有方差 $n^{-1}$ 的独立同分布中心化元素；$D$ 是与 $X$ 独立的对角随机矩阵，其元素独立同分布。该模型在统计学与机器学习中具有重要应用。本文在 $p$ 与 $n$ 同阶增长的假设下，证明 $Q$ 的极端边缘特征值渐近服从多种分布类型（取决于 $\Sigma$ 与 $D$），包括：Gumbel、Fr\'echet、Weibull、Tracy-Widom、高斯分布及其混合分布。一方面，当 $D$ 中随机变量具有无界支撑时，$Q$ 的边缘特征值渐近服从 Gumbel 或 Fr\'echet 分布（取决于 $D$ 的尾部衰减性质）；另一方面，当 $D$ 中随机变量具有有界支撑时，在关于 $\Sigma$ 的温和正则性假设下，$Q$ 的边缘特征值可呈现 Weibull、Tracy-Widom、高斯分布或其混合分布。基于理论结果，我们考虑两项重要应用：第一，针对椭圆分布数据，提出检测与估计可能存在的尖峰信号的统计量与流程；第二，在因子模型框架下，通过选取 $D$ 中的权重并采用乘子自助法，提出推断与估计因子数目的新算法。数值模拟验证了所提方法的准确性与有效性，且相比现有文献中的方法表现出更优性能。