About the asymptotic accuracy of Barron density estimates

By extending the information-theoretic arguments of previous papers dealing vith the Barren-type density estimates, and their consistency in information divergence and chi-square divergence, the problem of consistency in Csiszar's phi-divergence is motivated for general convex functions phi. The problem of consistency in phi-divergence is solved for all phi with phi(0) < infinity and phi(t) = O(t ill t) when t --> infinity. The problem of consistency in the expected phi-divergence is solved for all phi with t phi(1/t) + phi(t) = O(t(2)) when t --> infinity, Various stronger versions of these asymptotic restrictions are considered too. Assumptions about the model needed for the consistency are shown to depend on how strong these restrictions are.