In this paper we focus on the problem of estimating a bounded density using a finite combination of densities from a given class. We consider the Maximum Likelihood Estimator (MLE) and the greedy procedure described by Li and Barron (1999) under the additional assumption of boundedness of densities. We prove an $O\left(\frac{1}{\sqrt{n}}\right)$ bound on the estimation error which does not depend on the number of densities in the estimated combination. Under the boundedness assumption, this improves the bound of Li and Barron by removing the $logn$ factor and also generalizes it to the base classes with converging Dudley integral.

Classification: 62G05, 62G07, 62G20

Keywords: mixture density estimation, maximum likelihood, Rademacher processes

@article{PS_2005__9__220_0, author = {Rakhlin, Alexander and Panchenko, Dmitry and Mukherjee, Sayan}, title = {Risk bounds for mixture density estimation}, journal = {ESAIM: Probability and Statistics}, publisher = {EDP-Sciences}, volume = {9}, year = {2005}, pages = {220-229}, doi = {10.1051/ps:2005011}, zbl = {1141.62024}, mrnumber = {2148968}, language = {en}, url = {http://www.numdam.org/item/PS_2005__9__220_0} }

Rakhlin, Alexander; Panchenko, Dmitry; Mukherjee, Sayan. Risk bounds for mixture density estimation. ESAIM: Probability and Statistics, Volume 9 (2005) , pp. 220-229. doi : 10.1051/ps:2005011. http://www.numdam.org/item/PS_2005__9__220_0/

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