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Penalized maximum likelihood estimation with l1 penalty

J-M Loubes , Y. Yan


We focus on density estimation using penalized loglikelihood method. We aim at building an adaptive estimator in the sense that it converges at the optimal rate of convergence without prior knowledge of its regularity. For this, we penalize the loglikelihood by a function, which depends on the roughness of the density: the l1 norm of the wavelet coefficients of the logdensity. In this setting, we prove adaptivity for l2 norm over a certain class of sets, Besov spaces.


Density estimation, Penalized Maximum Likelihood, Complexity regularization, Wavelet Bases

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