We study improved approximations to the distribution of the largest eigenvalue ℓ ^ of the sample covariance matrix of n zero-mean Gaussian observations in dimension p + 1. We assume that one population principal component has variance ℓ > 1 and the remaining 'noise' components have common variance 1. In the high-dimensional limit p/n → γ > 0, we study Edgeworth corrections to the limiting Gaussian distribution of ℓ ^ in the supercritical case ℓ ​ > 1 + γ . The skewness correction involves a quadratic polynomial, as in classical settings, but the coefficients reflect the high-dimensional structure. The methods involve Edgeworth expansions for sums of independent non-identically distributed variates obtained by conditioning on the sample noise eigenvalues, and the limiting bulk properties and fluctuations of these noise eigenvalues.

中文翻译：

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尖峰 PCA 模型中最大特征值的 Edgeworth 校正

我们研究了 p + 1 维上 n 个零均值高斯观测的样本协方差矩阵的最大特征值 ℓ ^ 分布的改进近似值。我们假设一个总体主成分的方差 ℓ > 1 和其余的“噪声”成分具有共同方差 1. 在高维极限 p/n → γ > 0 中，我们研究了在超临界情况 ℓ > 1 + γ 下对 ℓ ^ 的极限高斯分布的 Edgeworth 修正。偏度校正涉及二次多项式，就像在经典设置中一样，但系数反映了高维结构。这些方法涉及通过对样本噪声特征值进行调节而获得的独立非相同分布变量的总和的埃奇沃斯展开，以及这些噪声特征值的限制体积特性和波动。