Let's make the problem as simple as possible. Assume two related random variables, $X_1$ and $X_2$. On the basis of some data we estimate their true means $\mu_{X_1}$ and $\mu_{X_2}$ by sample means $\hat\mu_{X_1}$ and $\hat\mu_{X_2}$. These estimates are unbiased.

But now let's sort our two random variables by their sample means and look at the variable with the highest sample mean. Now for this top-of-the-list random variable the sample mean is now a **biased** estimator of its true mean (under some reasonable assumptions, e.g. that the means of these random variables are themselves distributed in a certain way and that distribution has a mean) -- that's easy to verify by Monte-Carlo. For obviousness, take not two but a thousand random variables and make their true means similar.

The question is what is this bias and how do I analytically calculate it? I'd also appreciate some conceptual discussion on how does estimation bias arise out of sorting by estimated values.