Which model shall I use, and do fitting in “Two-stage Estimator”?

[jenifferYingyiWu]

The two-stage estimator is above.
After I have estimated the means and variances of errors.
How to implement the two-stage estimator by machine learning? Shall I separate the target items from training set and testing set?

If I separate them from train and test, then which model shall I use, and do fitting?

So, could you pls give me some suggestions and recommend some R codes?