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November 2017 Spectral analysis of high-dimensional sample covariance matrices with missing observations
Kamil Jurczak, Angelika Rohde
Bernoulli 23(4A): 2466-2532 (November 2017). DOI: 10.3150/16-BEJ815

Abstract

We study high-dimensional sample covariance matrices based on independent random vectors with missing coordinates. The presence of missing observations is common in modern applications such as climate studies or gene expression micro-arrays. A weak approximation on the spectral distribution in the “large dimension $d$ and large sample size $n$” asymptotics is derived for possibly different observation probabilities in the coordinates. The spectral distribution turns out to be strongly influenced by the missingness mechanism. In the null case under the missing at random scenario where each component is observed with the same probability $p$, the limiting spectral distribution is a Marčenko–Pastur law shifted by $(1-p)/p$ to the left. As $d/n\rightarrow y\in(0,1)$, the almost sure convergence of the extremal eigenvalues to the respective boundary points of the support of the limiting spectral distribution is proved, which are explicitly given in terms of $y$ and $p$. Eventually, the sample covariance matrix is positive definite if $p$ is larger than

\[1-(1-\sqrt{y})^{2},\] whereas this is not true any longer if $p$ is smaller than this quantity.

Citation

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Kamil Jurczak. Angelika Rohde. "Spectral analysis of high-dimensional sample covariance matrices with missing observations." Bernoulli 23 (4A) 2466 - 2532, November 2017. https://doi.org/10.3150/16-BEJ815

Information

Received: 1 October 2015; Published: November 2017
First available in Project Euclid: 9 May 2017

zbMATH: 06778247
MathSciNet: MR3648036
Digital Object Identifier: 10.3150/16-BEJ815

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 4A • November 2017
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