Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds

The classes of monotone or convex (and necessarily monotone) densities on     can be viewed as special cases of the classes of k - monotone densities on     . These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on     . In this paper we consider non-parametric maximum likelihood and least squares estimators of a k -monotone density g ₀. We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k ?1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives     , at a fixed point x ₀ under the assumption that     .