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.

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}, pages = {220--229}, publisher = {EDP-Sciences}, volume = {9}, year = {2005}, doi = {10.1051/ps:2005011}, zbl = {1141.62024}, mrnumber = {2148968}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2005011/} }

TY - JOUR AU - Rakhlin, Alexander AU - Panchenko, Dmitry AU - Mukherjee, Sayan TI - Risk bounds for mixture density estimation JO - ESAIM: Probability and Statistics PY - 2005 SP - 220 EP - 229 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2005011/ DO - 10.1051/ps:2005011 LA - en ID - PS_2005__9__220_0 ER -

%0 Journal Article %A Rakhlin, Alexander %A Panchenko, Dmitry %A Mukherjee, Sayan %T Risk bounds for mixture density estimation %J ESAIM: Probability and Statistics %D 2005 %P 220-229 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2005011/ %R 10.1051/ps:2005011 %G en %F 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/articles/10.1051/ps:2005011/

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