Best Asymptotic Normality of the Kernel Density Entropy Estimator for Smooth Densities

In the random sampling setting we estimate the entropy of a probability density distribution by the entropy of a kernel density estimator using the double exponential kernel. Under mild smoothness and moment conditions we show that the entropy of the kernel density estimator equals a sum of independent and identically distributed (i.i.d.) random variables plus a perturbation which is asymptotically negligible compared to the parametric rate n 1=2. An essential part in the proof is obtained by exhibiting almost sure bounds for the Kullback–Leibler divergence between the kernel density estimator and its expected value. The basic technical tools are Doob’s submartingale inequality and convexity (Jensen’s inequality).