On the Kozachenko-Leonenko entropy estimator

We study in details the bias and variance of the entropy estimator proposed by Kozachenko and Leonenko for a large class of densities on $\mathbb{R}^d$. We then use the work of Bickel and Breiman to prove a central limit theorem in dimensions $1$ and $2$. In higher dimensions, we provide a development of the bias in terms of powers of $N^{-2/d}$. This allows us to use a Richardson extrapolation to build, in any dimension, an estimator satisfying a central limit theorem and for which we can give some some explicit (asymptotic) confidence intervals.