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Consider the problem of estimating the Shannon entropy of a distribution over$k$ elements from $n$ independent samples. We show that the minimax mean-squareerror is within universal multiplicative constant factors of $$\Big(\frac{k }{n\log k}\Big)^2 + \frac{\log^2 k}{n}$$ if $n$ exceeds a constant factor of$\frac{k}{\log k}$; otherwise there exists no consistent estimator. Thisrefines the recent result of Valiant-Valiant \cite{VV11} that the minimalsample size for consistent entropy estimation scales according to$\Theta(\frac{k}{\log k})$. The apparatus of best polynomial approximationplays a key role in both the construction of optimal estimators and, via aduality argument, the minimax lower bound.

Minimax rates of entropy estimation on large alphabets via best polynomial approximation