  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.
