Risk bounds for mixture density estimation
ESAIM: Probability and Statistics, Volume 9  (2005), p. 220-229

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(1 n) 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.

DOI : https://doi.org/10.1051/ps:2005011
Classification:  62G05,  62G07,  62G20
Keywords: mixture density estimation, maximum likelihood, Rademacher processes
     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/

[1] A.R. Barron, Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39 (1993) 930-945. | Zbl 0818.68126

[2] A.R. Barron, Approximation and estimation bounds for artificial neural networks. Machine Learning 14 (1994) 115-133. | Zbl 0818.68127

[3] L. Birgé and P. Massart, Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 (1993) 113-150. | Zbl 0805.62037

[4] R.M. Dudley, Uniform Central Limit Theorems. Cambridge University Press (1999). | MR 1720712 | Zbl 0951.60033

[5] L.K. Jones, A simple lemma on greedy approximation in Hilbert space and convergence rates for Projection Pursuit Regression and neural network training. Ann. Stat. 20 (1992) 608-613. | Zbl 0746.62060

[6] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, New York (1991). | MR 1102015 | Zbl 0748.60004

[7] J. Li and A. Barron, Mixture density estimation, in Advances in Neural information processings systems 12, S.A. Solla, T.K. Leen and K.-R. Muller Ed. San Mateo, CA. Morgan Kaufmann Publishers (1999).

[8] J. Li, Estimation of Mixture Models. Ph.D. Thesis, The Department of Statistics. Yale University (1999).

[9] C. Mcdiarmid, On the method of bounded differences. Surveys in Combinatorics (1989) 148-188. | Zbl 0712.05012

[10] S. Mendelson, On the size of convex hulls of small sets. J. Machine Learning Research 2 (2001) 1-18. | Zbl 1008.68107

[11] P. Niyogi and F. Girosi, Generalization bounds for function approximation from scattered noisy data. Adv. Comput. Math. 10 (1999) 51-80. | Zbl 1053.65506

[12] S.A. Van De Geer, Rates of convergence for the maximum likelihood estimator in mixture models. Nonparametric Statistics 6 (1996) 293-310. | Zbl 0872.62039

[13] S.A. Van De Geer, Empirical Processes in M-Estimation. Cambridge University Press (2000).

[14] A.W. Van Der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes with Applications to Statistics. Springer-Verlag, New York (1996). | MR 1385671 | Zbl 0862.60002

[15] W.H. Wong and X. Shen, Probability inequalities for likelihood ratios and convergence rates for sieve mles. Ann. Stat. 23 (1995) 339-362. | Zbl 0829.62002