Optimal entropy estimation on large alphabets via best polynomial approximation
Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 2015

Consider the problem of estimating the Shannon  entropy of a distribution on $ k$  elements from $ n$  independent  samples. We show that the minimax mean-square error is within  universal multiplicative constant factors of  $\left( \frac{n}{k \log n} \right)^{2} + \frac{\log^2 k}{n}$. This  implies the recent result of Valiant-Valiant [ 1 ] that the minimal  sample size for consistent entropy estimation scales according to $\Theta( \frac{k}{\log k} )$.  The apparatus of best polynomial approximation plays a key role in both the minimax lower bound and the construction of optimal estimators.