Matrix completion based on low-rank models is very popular and comes with powerful algorithms and theoretical guarantees. However, existing methods do not consider the case of values missing not at random (MNAR) which are widely encountered in practice. Considering a data matrix generated from a probabilistic principal component analysis (PPCA) model containing several MNAR variables, we propose estimators for the means, variances and covariances related to the MNAR missing variables and study their consistency. The proposed estimators present the advantage of being computed without explicitly modeling the MNAR mechanism and by only using observed data. In addition, we propose an imputation method of the data matrix and an estimation of the PPCA loading matrix. We compare our proposal with the classical methods used in low-rank models, as iterative methods based on singular value decomposition.
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