Maximum smoothed likelihood density estimation

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Abstract
We interpret the kernel estimator for the density as the solution to a maximum smoothed likelihood problem, and show under suitable conditions that it converges to the true densityin the sense that the Kullback-Leibler information number of the estimator relative to the true density converges to 0 in probability. We also consider the convergence in probability of the naiveestimator for the entropy number of the density. The conditions under which all this happens say essentially that the true density as well as the smoothed true density have finite entropy. The significance of the maximum (smoothed) likelihood set-up is that the relevant random variables are nonnegative, and all we have to do is show that their expected values converge to 0.

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