Conditional Density Estimation by Penalized Likelihood Model Selection

In this paper, we consider conditional density estimation, and propose a general condition on the penalty of a penalized maximum likelihood estimate to obtain oracle type inequality with Kullback-Leibler type loss. Our aim is threefold: to extend a model selection theorem obtained by Massart for density estimation, to illustrate this theorem with families of "piecewise constant" conditional density estimator, and to provide some theoretical justification for a companion paper on unsupervised segmentation based on spatially varying Gaussian mixture estimation.