Distribution Estimates Consistent in χ²-Divergence

Abstract

For a class of histogram based distribution estimators it is proved the consistency in χ2-divergence and expected χ2-divergence. For one of these estimators, introduced formerly by Barron, is also evaluated the rate of consistency in the expected χ2-divergence. These results are stronger than similar results formerly established in the literature for the total variation and Kullback's discrimination information, which are divergences dominated by the χ2-divergence.