Calculating a covariance matrix with missing data

[TravisJay]

Consider a co-variance matrix A such 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. Let's 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.

 

[bahamagreen]

See if non-parametric statistical analysis might be what you are needing...

 

[Stephen Tashi]

Consider a co-variance matrix A such 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.

The way such a matrix is computed is from the joint distribution of $X_i, X_j$. It isn't computed from sample data.

Consider the case that each variable X has a different sample size.

Apparently, what you want to do is estimate the covariance matrix.

You should look up methods of estimating covariance from samples that have missing data.

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.

You haven't given enough information to define the case. There is no general "best" method for doing this unless you make some assumptions - for example, assumptions about what family of distributions generated the data.

http://icml.cc/discuss/2012/313.html