Effective construction of modified histograms in higher dimensions

Abstract

Density estimation raises delicate problems in higher dimensions especially when strong convergence is required and data marginals can be highly correlated. Modified histograms have been introduced to circumvent the problem of low bin counts when convergence is considered in the sense of information divergence. These estimates are defined from some reference probability density and an associated partition which is defined in the univariate case fromni the quantiles of the reference density. Therefore, in the multivariate case, the definition of the partition causes an additional probleni related to the lack of total order. In this paper, we present a method for constructing modified multivariate histograms such that the corresponding partition is well adapted to the observed data. The approach is based on a data-driven coordinate system selected by cross-validation. We discuss the performance of our estimate with the help of a finite sample sirnulation study. © 2005 Springer Science+Business Media, Inc.