Densities with optimal smoothness in moment problems
Abstract
Moment problem has been studied for a long time by many authors and numerous results are known regarding finite discrete distribution with prescribed moments. Finding smoothest function with various properties has recently been a popular topic in both mathematics and statistics. In function fitting problems, some difficulties arise from the given constraints. If there are finite number of constraints, the problem can be treated quite easily. Nonparametric regression problems usually have a finite number of constraints. But in density fitting problems, a serious difficulty comes in by the nature of the problem, the nonnegativity. This constraint is actually an infinite dimensional constraint. In almost all of the work done in density estimation problems, this difficulty was avoided by considering log density or a function of which square gives a density. In this thesis we consider finding densities with optimal smoothness under the moment conditions and nonnegativity constraint. The space of functions considered is the Sobolev space $W\sbsp{m}{2}$ (0,1) and the target functional to be minimized is the seminorm $\Vert f\sp{(m)}\Vert\sb{L\sp2}$, which measures the roughness of the function f. The existence of the minimizer is shown and usable necessary and sufficient conditions characterizing the minimizer are found. The theorems for the characterization of the minimizer heavily rely on the theory of optimal control. Some examples are given in finding the smoothest density with all the constraints, where the theorems for the characterization are used. Finally discrete approximations to the solution and an application to density estimation are studied.
Degree
Ph.D.
Advisors
Studden, Purdue University.
Subject Area
Statistics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.