equal covariance matrices

Question

Let's pretend that there exist generators which generate datasets, is it safe to assume that the 2 datasets were generated by the same generator if they present equal covariance structures? In other words, What can I say about 2 datasets whose covariance matrices are identical?

Answer 1

You can say that they have identical covariance matrices. It may sounds stupid, but it is not. Almost any other statement depends on your assumptions. Namely, if you are assuming a (family) of generating models, let us say, a normal distribution and if the data fits well the distribution (using K-S test) and by combining these two data sets the estimators of the parameters haven't changed much - then you may infer with some confidence (probability) that both data sets have the same generating model. However, without assuming nothing or any other prior knowledge - you cannot say much.