L2-density estimation under constraints

In this paper, we are interested in non parametric density estimation under constraints. It generalises a previous paper which was devoted to density estimation with non-positive kernels. The resulting density approximation improves the estimation (by reducing the bias) but provides negative values. Therefore, we have proposed a projection method on the space of probability densities and an algorithm designed to generate a sample from the projected density. We present here a generalization of this work in considering several linear constraints on the estimated density. These constraints represent an a priori knowledge of the underlying density. For example, the support, some moments or quantiles of the approximated density can be set a priori by the user. We prove that the projected density on the closed and convex set of functions satisfying some the constraints has a simple and explicit form. Some simulations show that the proposed solution outperforms alternative solutions proposed in the literature.