Estimating population means after a selection procedure The problem is somewhat related to sequential procedures (e.g. Paulson; Gupta and Miescke)
I have a hundred engines generating normal random numbers with true (unknown) means $E_1,E_2,E_3...E_{100}$, each having the same variance, of say V.
After 50 samples have been collected from each of the engines, there is a forced selection of the 10 engines with the highest sample means, irrespective of our confidence in the means.
Let these sample means be $X^1, X^2, X^3...X^{10}$
After this selection procedure, another 50 samples, each, are collected from the selected 10 engines. Let the sample means of these new samples be $Y^1, Y^2, Y^3...Y^{10}$
What is my unbiased estimate (with least standard error) of the true means of these 10 engines

Those values $X^1, \ldots, X^{10}$ from before the selection should not be used for the computation of their means. They depend on the values of all the 100 engines and thus suffer from selection bias: they will have the tendency to be unusually large.
And since you don't know the means of all the engines, you also cannot figure out what amount of information inside the preselection measurements you could still use for variance reduction in the computation of the true mean of the ten selected engines (since you mentioned that you are looking for an MVUE).
