Consider a co-variance matrix(adsbygoogle = window.adsbygoogle || []).push({}); Asuch that each element a_{i,j}= E(X_{i}X_{j}) - E(X_{i}) E(X_{j}) where X_{i},X_{j}are random variables.

Consider the case that each variable X has a different sample size. Lets say that X_{i}contains the elements x_{i,1}, …, x_{i,N}, and X_{j}contains the elements x_{j,1}, ..., x_{j,n}where each element is paired up to element n and N > n.

In this case, for each covariance a_{i,j}, is it acceptable to trim the sample size for each X_{i}and X_{j}to n and continue the calculation? (I'm not sure if trim is the correct terminology but it seems to meet my needs).

If it is acceptable to trim, then is it necessary to trim to the smallest n of all of the random variables X, or can I just trim to the smallest of the pair?

I'd appreciate it if anyone can point me in the direction of some literature that explains this in detail. I've been struggling to find something that is specific to this case.

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# Calculating a covariance matrix with missing data

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