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.

翻译：我们考虑基于样本数据估计概率密度函数的问题，利用来自某个分量类别的有限混合密度。为此，我们引入$h$-提升的Kullback-Leibler（KL）散度作为标准KL散度的推广，并将其作为实施风险最小化的准则。在紧支撑假设下，我们证明了使用$h$-提升的KL散度时，期望估计误差具有$\mc{O}(1/{\sqrt{n}})$的界，这一结果将Rakhlin等人（2005，ESAIM：概率与统计，第9卷）以及Li和Barron（1999，神经信息处理系统进展，第12卷）的研究推广到允许对非严格正密度函数进行风险界定的情形。我们利用Majorization-Maximization框架开发了相应最大$h$-提升似然估计（$h$-MLLEs）的计算方法，并提供了支持我们理论界的实验结果。