Estimation of a K − Monotone Density , Part 1 : Characterizations , Consistency , and Minimax Lower

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 (−1)g is nonnegative, nonincreasing and convex for l = 0, . . . , k−2 if k ≥ 2, and g is simply nonincreasing if k = 1. These classes of shaped 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 (∞−monotone) densities. It is well-known that a density is completely monotone if and only if it is a scale mixture of exponential densities (Bernstein’s theorem). Thus one motivation for studying the problem of estimation of a k−monotone density is to try to gain insight into the problem of estimating a completely monotone density. In this series of four papers we consider both (nonparametric) Maximum Likelihood estimators and Least Squares estimators of a k−monotone estimator. 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 g0(x0) ∗Research supported in part by National Science Foundation grant DMS-0203320 †Research supported in part by National Science Foundation grant DMS-0203320, NIAID grant 2R01 AI291968-04, and an NWO Grant to the Vrije Universiteit, Amsterdam ‡Corresponding author AMS 2000 subject classifications: Primary 62G05, 60G99; secondary 60G15, 62E20