Spectral analysis of high-dimensional sample covariance matrices with missing observations

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 Mar\v{c}enko-Pastur law shifted by $(1-p)/p$ to the left. As $d/n\rightarrow y< 1$, 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 $$ 1-\left(1-\sqrt{y}\right)^2, $$ whereas this is not true any longer if $p$ is smaller than this quantity.