MULTIVARIATE HISTOGRAMS WITH DATA-DEPENDENT PARTITIONS

We consider estimation of multivariate densities with histograms which are based on data-dependent partitions. We find data-dependent partitions by minimizing a complexity-penalized error criterion. The estimator may also be characterized as a series estimator whose basis is chosen empirically. We show that the estimator achieves minimax rates of convergence tip to a logarithmic factor over a scale of smoothness classes containing functions with anisotropic and spatially varying smoothness. The method may also be viewed as based on the presmoothing of data. We show how the optimal amount of presmoothing depends on the spatial inhomogeneity of the density.