I think the article by Richard Samworth lays out the paradox better. The whole article is worth a read, but here's the paradox part " To give an unusual example

I think the article by Richard Samworth lays out the paradox better. The whole article is worth a read, but here's the paradox part " To give an unusual example to emphasise the point, suppose that we were interested in estimating the proportion of the US electorate who will vote for Barack Obama, the proportion of babies born in China that are girls and the proportion of Britons with light-coloured eyes. Then our James–Stein estimate of the proportion of democratic voters depends on our hospital and eye colour data! " <http://www.statslab.cam.ac.uk/~rjs57/SteinParadox.pdf> http://www.statslab.cam.ac.uk/~rjs57/SteinParadox.pdf> Surely that's paradoxical! The OP's post is an outstanding exposition of James-Stein estimators though, so thanks for the post. There seems to be lots of connection between these and doing linear regressions with regularisation in machine learning.