Nonparametric estimation of a k-monotone density: A new asymptotic distribution theory.

In this dissertation, we consider the problem of nonparametric estimation of a k-monotone density on (0,∞) for a fixed integer k > 0 via the methods of Maximum Likelihood (ML) and Least Squares (LS). In the introduction, we present the original question that motivated us to look into this problem and also put other existing results in our general framework. In Chapter 2, we study the MLE and LSE of a k-monotone density based on n i.i.d. observations. Here, our study of the estimation problem is local in the sense that we only study the estimator and its derivatives at a fixed point x_0. Under some specific working assumptions, asymptotic minimax lower bounds for estimating g^{(j)}_0 (x_0), j=1,...,k-1, are derived. These bounds show that the rates of convergence of any estimator of g^{(j)}_0 (x_0) can be at most n^{-(k-j)/(2k+1)}. Furthermore, under the same working assumptions we prove that this rate is achieved by the j-th derivative of either the MLE or LSE if a certain conjecture concerning the error in a particular Hermite interpolation problem holds. To make the asymptotic distribution theory complete, the limiting distribution needs to be determined. This distribution depends on a very special stochastic process H_k which is almost surely uniquely defined on R. Chapter 3 is essentially devoted to an effort to prove the existence of such a process and to establish conditions characterizing it. It turns out that we can establish the existence and uniqueness of the process Hk if the same conjecture mentioned above with the finite sample problem holds. If Y_k is the (k-1) -fold integral of two-sided Brownian motion + k!/(2k)! t2k, then H_k is a random spline of degree 2k-1 that stays above Y_k if k is even and below it if k is odd. By applying a change of scale, our results include the special cases of estimation of monotone densities (k =1), and monotone and convex densities (k =2) for which an asymptotic distribution theory is available. Iterative spline algorithms developed to calculate the estimators and approximate the process H_k on finite intervals are described in Chapter 4. These algorithms exploit both the spline structure of the estimators and the process H_k as well as their characterizations and are based on iterative addition and deletion of the knot points.