期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2017
卷号：79
期号：2
页码：254-297
DOI：10.1007/s13171-017-0106-6
语种：English
出版社：Indian Statistical Institute
摘要：Let X be a mean zero Gaussian random vector in a separable Hilbert space \({\mathbb H}\) with covariance operator \({\Sigma }:={\mathbb E}(X\otimes X).\) Let \({\Sigma }={\sum }_{r\geq 1}\mu _{r} P_{r}\) be the spectral decomposition of Σ with distinct eigenvalues \(\mu _{1}>\mu _{2}> \dots \) and the corresponding spectral projectors \(P_{1}, P_{2}, \dots .\) Given a sample \(X_{1},\dots , X_{n}\) of size n of i.i.d. copies of X , the sample covariance operator is defined as \(\hat {\Sigma }_{n} := n^{-1}{\sum }_{j=1}^{n} X_{j}\otimes X_{j}.\) The main goal of principal component analysis is to estimate spectral projectors \(P_{1}, P_{2}, \dots \) by their empirical counterparts \(\hat P_{1}, \hat P_{2}, \dots \) properly defined in terms of spectral decomposition of the sample covariance operator \(\hat {\Sigma }_{n}.\) The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic \(\|\hat P_{r}-P_{r}\|_{2}^{2},\) where \(\|\cdot \|_{2}^{2}\) is the squared Hilbert–Schmidt norm. This is done in a “high-complexity” asymptotic framework in which the so called effective rank \(\textbf {r}({\Sigma }):=\frac {\text {tr}({\Sigma })}{\|{\Sigma }\|_{\infty }}\) (tr(⋅) being the trace and \(\|\cdot \|_{\infty }\) being the operator norm) of the true covariance Σ is becoming large simultaneously with the sample size n , but r (Σ) = o ( n ) as \(n\to \infty .\) In this setting, we prove that, in the case of one-dimensional spectral projector P r , the properly centered and normalized statistic \(\|\hat P_{r}-P_{r}\|_{2}^{2}\) with data-dependent centering and normalization converges in distribution to a Cauchy type limit. The proofs of this and other related results rely on perturbation analysis and Gaussian concentration.
关键词：Sample covariance ; Spectral projectors ; Effective rank ; Principal component analysis ; Asymptotic distribution ; Perturbation theory
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