Approximation and Estimation of s-Concave Densities via R\'enyi Divergences

Understand

In this paper, we study the approximation and estimation of $s$-concave densities via R\'enyi divergence.

• We first show that the approximation of a probability measure $Q$ by an $s$-concave densities exists and is unique via the procedure of minimizing a divergence functional proposed by Koenker and Mizera (2010) if and only if $Q$ admits full-dimensional support and a first moment.
• We also show continuity of the divergence functional in $Q$: if $Q_n \to Q$ in the Wasserstein metric, then the projected densities converge in weighted $L_1$ metrics and uniformly on closed subsets of the continuity set of the limit.
• Moreover, directional derivatives of the projected densities also enjoy local uniform convergence.