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**Spectral analysis of high-dimensional sample covariance matrices with missing observations.**
*(English)*
Zbl 1388.62152

Summary: 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 [V. A. Marchenko and L. A. Pastur, Math. USSR, Sb. 1, 457–483 (1968; Zbl 0162.22501); Mat. Sb., Nov. Ser. 72(114), 507–536 (1967; Zbl 0152.16101)] 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.

### MSC:

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

60B10 | Convergence of probability measures |

60B20 | Random matrices (probabilistic aspects) |

62P12 | Applications of statistics to environmental and related topics |