A normal limit theorem for moment sequences, rigid designs and model selection for polynomial regression

Fu-Chuen Chang, Purdue University

Abstract

First part. Let $\Lambda$ denote the set of probability measures on (0,1) and let $c\sb{k} = \int\sbsp{0}{1} x\sp{k}d\lambda,k = 1,2, \cdots$ be the ordinary moments. Let $M\sb{n} = \{(c\sb1, \cdots, c\sb{n})\vert\lambda \in \Lambda\}$ and consider a uniform probability measure $P\sb{n}$ on the set $M\sb{n}$. We show that as $n \to \infty$, the fixed section $(c\sb1, \cdots, c\sb{k})$ properly normalized, is asymptotically normally distributed. That is, $\sqrt{n}\lbrack(c\sb1, \cdots, c\sb{k}) - (c\sbsp{1}{0}, \cdots, c\sbsp{k}{0})\rbrack$ converges to $MVN(0,\Sigma)$ where $c\sbsp{i}{0}$ correspond to the arc-sin law. Some properties of the $k \times k$ matrix $\Sigma$ and some further discussion is given. Second part. Rigid design is defined and investigated. It is equivalent to the so-called Lebesgue problem in interpolation theory. For one-dimensional polynomial regressional models, we derive an iterative algorithm based on the Newton-Ralphson method for nonlinear systems to compute the rigid design points and the so-called optimal Lebesgue constant. For the two-dimensional case, we consider polynomial regression models with three terms only, constant and two linear terms, and a compact convex design space. Necessary conditions for rigid designs are derived. Product type polynomial regression models and design spaces is also studied. Third part. Model selection for polynomial regression is considered. The D-optimal criterion is employed to select the robust model. Several examples are studied. In two dimensions, we consider four regression functions 1, x, y, $\lambda\sb1 x\sp2$ + $\lambda\sb{2}y\sp2$ + $\lambda\sb3 xy$ and a design point set with four distinct and arbitrary points in $R\sp2$. We derive the solution set of $\lambda\sb1, \lambda\sb2, \lambda\sb3$ such that the design with uniform weight on the four design points is D-optimal over the convex hull of four design points. Two special designs and models in $R\sp2$ are generalized to d dimensions.

Degree

Ph.D.

Advisors

Studden, Purdue University.

Subject Area

Statistics

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