We consider the class of Erlang mixtures for the task of density estimation on the positive real line when the only available information is given as local moments, a histogram with potentially higher order moments in some bins. By construction, the obtained moment problem is ill-posed and requires regularization. Several penalties can be used for such a task, such as a lasso penalty for sparsity of the representation, but we focus here on a simplified roughness penalty from the P-splines literature. We show that the corresponding hyperparameter can be selected without cross-validation through the computation of the so-called effective dimension of the estimator, which makes the estimator practical and adapted to these summarized information settings. The flexibility of the local moments representations allows interesting additions such as the enforcement of Value-at-Risk and Tail Value-at-Risk constraints on the resulting estimator, making the procedure suitable for the estimation of heavy-tailed densities.

翻译：我们考虑在仅能获取局部矩信息（即某些区间内可能包含高阶矩的直方图）的情况下，针对正实数轴上的密度估计任务采用Erlang混合模型类。由于构造方式所限，该矩问题具有不适定性，需要正则化处理。虽然可采用多种惩罚函数完成此任务（如用于表示稀疏性的lasso惩罚），但本文聚焦于P样条文献中的简化粗糙度惩罚。研究表明，通过计算所谓估计量有效维度可在无需交叉验证的情况下选择相应的超参数，这使得该估计量具有实用价值并适用于此类汇总信息场景。局部矩表示的灵活性允许在所得估计量上施加诸如风险价值（VaR）与尾部风险价值（TVaR）约束等附加条件，从而使该方法适用于重尾密度估计。