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Best asymptotic normality of the kernel density entropy estimator for smooth densities

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2 Author(s)
P. B. Eggermont ; Dept. of Math. Sci., Delaware Univ., Newark, DE, USA ; V. N. LaRiccia

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)

Published in:

IEEE Transactions on Information Theory  (Volume:45 ,  Issue: 4 )