Alain Berlinet
University of Montpellier
France
Modified histograms and dynamical systems
Modified histograms introduced by Barron, Gyorfi and van der Meulen have recently been shown to have nice properties with respect to the Kullback-Leibler information criterion. Some of these properties will be reviewed: convergence, asymptotic normality, optimization of the Kullback-Leibler cross-validation criterion.
Since these density estimates are obtained by combining an initial reference density to the observed data, the procedure can be seen as a dynamical system in a functional space with the reference density as initial state. I will prove under mild assumptions that the trajectory is almost surely stationary. This opens the way to simultaneous optimization of the reference density and the number of bins.