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EDGEWORTH CORRECTION FOR THE LARGEST EIGENVALUE IN A SPIKED PCA MODEL.

ABSTRACT: We study improved approximations to the distribution of the largest eigenvalue ?^ of the sample covariance matrix of n zero-mean Gaussian observations in dimension p + 1. We assume that one population principal component has variance ? > 1 and the remaining 'noise' components have common variance 1. In the high-dimensional limit p/n ? ? > 0, we study Edgeworth corrections to the limiting Gaussian distribution of ?^ in the supercritical case ??>?1?+? . 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.

SUBMITTER: Yang J

PROVIDER: S-EPMC6420228 | biostudies-literature | 2018 Oct

REPOSITORIES: biostudies-literature

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EDGEWORTH CORRECTION FOR THE LARGEST EIGENVALUE IN A SPIKED PCA MODEL.

Yang Jeha J Johnstone Iain M IM

Statistica Sinica 20181001 4

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 skewn ...[more]

PMID: 30886511

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