  We consider the problem of estimating probability density functions based on
sample data, using a finite mixture of densities from some component class. To
this end, we introduce the $h$-lifted Kullback--Leibler (KL) divergence as a
generalization of the standard KL divergence and a criterion for conducting
risk minimization. Under a compact support assumption, we prove an
$\mc{O}(1/{\sqrt{n}})$ bound on the expected estimation error when using the
$h$-lifted KL divergence, which extends the results of Rakhlin et al. (2005,
ESAIM: Probability and Statistics, Vol. 9) and Li and Barron (1999, Advances in
Neural Information ProcessingSystems, Vol. 12) to permit the risk bounding of
density functions that are not strictly positive. We develop a procedure for
the computation of the corresponding maximum $h$-lifted likelihood estimators
($h$-MLLEs) using the Majorization-Maximization framework and provide
experimental results in support of our theoretical bounds.
