JSM 2005 Online Program

Abstract #302295

This is the preliminary program for the 2005 Joint Statistical Meetings in Minneapolis, Minnesota. Currently included in this program is the "technical" program, schedule of invited, topic contributed, regular contributed and poster sessions; Continuing Education courses (August 7-10, 2005); and Committee and Business Meetings. This on-line program will be updated frequently to reflect the most current revisions.

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Activity Number:	205
Type:	Invited
Date/Time:	Tuesday, August 9, 2005 : 8:30 AM to 10:20 AM
Sponsor:	IMS
Abstract - #302295
Title:	Nonparametric Estimation of a k-monotone Density: Asymptotic Distribution Theory
Author(s):	Fadoua Balabadaoui*+
Companies:	Institut fuer Mathematische Stochastik
Address:	Maschmuehlenweg 8-10, Goettingen, , Germany
Keywords:	k-monotone ; completely monotone ; mixture model ; nonparametric estimation ; asymptotic distribution ; Hermite interpolation
Abstract:	In this talk, we consider the problem of nonparametric estimation of a k-monotone density at a point x for a fixed integer k via the methods of Maximum Likelihood (ML) and Least Squares (LS). We present the original question that motivated us to investigate this problem, and also put other existing results in our general framework. We study the MLE and LSE of a k-monotone density g and its derivatives up to order k-1 based on n i.i.d. observations. Under some specific working assumptions, asymptotic minimax lower bounds for estimating the derivatives are derived. These bounds show the rates of convergence of any estimator of the j-th derivative of g 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. If the same conjecture is true, we show that the asymptotic distribution depends on a random spline of degree 2k-1 that stays above (below) the (k-1)-fold integral of two-sided Brownian motion plus a deterministic trend if k is even (odd).

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