What is the procedure for estimating the population covariance matrix from many incomplete sample covariance matrices?

For example, say my data consist of 7 flavors of time series, A-G. My goal is to estimate the population covariance among all 7 flavors. However, each of my (many) observations and thus my sample covariance matrices only contain 4 flavors each, selected at random, e.g. [B C F G], [A C D F].

My intuition is that it is some variant of mixed-effect modeling with pair-wise covariance being the fixed effect and per-observation flavor grouping being the random effect.



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