Missing data present an important challenge when dealing with high dimensional data arranged in the form of an array. In this paper, we propose methods for estimation of the parameters of array variate normal probability model from partially observed multi-way data. The methods developed here are useful for missing data imputation, estimation of mean and covariance parameters for multi-way data. A multi-way semi-parametric mixed effects model that allows separation of multi-way covariance effects is also defined and an efficient algorithm for estimation based on the spectral decompositions of the covariance parameters is recommended. We demonstrate our methods with simulations and with real life data involving the estimation of genotype and environment interaction effects on possibly correlated traits.