Estimation of a k-monotone density, part 1: characterizations consistency, and minimax lower bounds
Shape constrained densities are encountered in many nonparametric estimation problems. The classes of monotone or convex (and monotone) densities can be viewed as special cases of the classes of k-monotone densities. A density g is said to be k-monotone if its successive derivatives up to order k-2 are alternately negative or positive, with the k-2 derivative convex (or concave) according as k is even (or odd) respectively. g is simply nonincreasing if k=1. These classes of shape constrained densities 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 densities with k infinite. In this series of four papers we consider both (nonparametric) Maximum Likelihood estimators and Least Squares estimators of a k-monotone density. In this first part (part 1), we prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of order k (degree k-1) with simple knots. We further provide asymptotic minimax risk lower bounds for estimating a k-monotone density g_0 (x_0) and its derivatives g_0^{(j)}(x_0), for j=1, ..., k-1, at a fixed point x_0 under the assumption that the k-th derivative of g_0 at x_0 is non-zero.