Shape constraints play an increasingly prominent role in nonparametric function estimation. While considerable recent attention has been focused on log concavity as a regularizing device in nonparametric density estimation, weaker forms of concavity constraints encompassing larger classes of densities have received less attention but offer some additional flexibility. Heavier tail behavior and sharper modal peaks are better adapted to such weaker concavity constraints. When paired with

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... te maximal entropy estimation criteria these weaker constraints yield tractable, convex optimization problems that broaden the scope of shape constrained density estimation in a variety of applied subject areas. In contrast to our prior work, Koenker and Mizera (2010), that focused on the log concave (α = 1) and Hellinger (α = 1/2) constraints, here we describe methods enabling imposition of even weaker, α ≤ 0 constraints. An alternative formulation of the concavity constraints for densities in dimension d ≥ 2 also significantly expands the applicability of our proposed methods for multivariate data. Finally, we illustrate the use of the Renyi divergence criterion for norm-constrained estimation of densities in the absence of a shape constraint.