  We study the problem of testing whether the missing values of a potentially
high-dimensional dataset are Missing Completely at Random (MCAR). We relax the
problem of testing MCAR to the problem of testing the compatibility of a
sequence of covariance matrices, motivated by the fact that this procedure is
feasible when the dimension grows with the sample size. Tests of compatibility
can be used to test the feasibility of positive semi-definite matrix completion
problems with noisy observations, and thus our results may be of independent
interest. Our first contributions are to define a natural measure of the
incompatibility of a sequence of correlation matrices, which can be
characterised as the optimal value of a Semi-definite Programming (SDP)
problem, and to establish a key duality result allowing its practical
computation and interpretation. By studying the concentration properties of the
natural plug-in estimator of this measure, we introduce novel hypothesis tests
that we prove have power against all distributions with incompatible covariance
matrices. The choice of critical values for our tests rely on a new
concentration inequality for the Pearson sample correlation matrix, which may
be of interest more widely. By considering key examples of missingness
structures, we demonstrate that our procedures are minimax rate optimal in
certain cases. We further validate our methodology with numerical simulations
that provide evidence of validity and power, even when data are heavy tailed.
