Shape Constrained Density Estimation via Penalized Rényi Divergence

Abstract. 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 appropriate 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.