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EXCHANGEABLE PAIRS IN STEIN'S METHOD OF DISTRIBUTIONAL
APPROXIMATION
by
Nathan Forrest Ross
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2009
Copyright 2009 Nathan Forrest Ross
Object Description
| Title | Exchangeable pairs in Stein's method of distributional approximation |
| Author | Ross, Nathan Forrest |
| Author email | nathanfr@usc.edu; ross@stat.berkeley.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Mathematics |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2009-06-18 |
| Date submitted | 2009 |
| Restricted until | Unrestricted |
| Date published | 2009-08-06 |
| Advisor (committee chair) | Fulman, Jason |
| Advisor (committee member) |
Goldstein, Larry M. Kempe, David |
| Abstract | Stein's method is a powerful tool used to obtain error terms in distributional approximation and limit theorems. In one formulation, an auxiliary stochastic object termed an exchangeable pair must be created in order to apply the method. This dissertation has two main purposes, the first is to examine the effect that the choice of exchangeable pair has in the computation and quality of the error, and the second is to derive new tools using exchangeable pairs and the main principles of Stein's method.; In examining the method, we will focus on examples of a combinatorial and algebraic nature in theorems pertaining to normal and Poisson approximation. For normal approximation, we analyze the role of the exchangeable pair for the binomial distribution and the Plancherel measure of a random walk on the symmetric group. Poisson approximation for exchangeable pairs is less developed than normal approximation, so that the examples presented are the binomial and negative binomial distributions, both of which have representations as sums of independent random variables.; The new tools developed here include exchangeable pair formulations for Poisson and translated Poisson approximation which are more in the spirit of existing normal approximation theorems than those available. Also, there have been many recent developments adapting Stein's method to approximation by discrete analogs of the normal distribution which contain a smoothing term that is the main technical difficulty in applications. Exchangeable pairs can be used to derive a simple bound on this term that easily reproduces many results found in the literature. |
| Keyword | exchangeable pairs; Stein's method |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m2509 |
| Contributing entity | University of Southern California |
| Rights | Ross, Nathan Forrest |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@dots.usc.edu |
| Filename | etd-Ross-3031 |
| Archival file | uscthesesreloadpub_Volume32/etd-Ross-3031.pdf |
Description
| Title | Page 1 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@dots.usc.edu |
| Full text | EXCHANGEABLE PAIRS IN STEIN'S METHOD OF DISTRIBUTIONAL APPROXIMATION by Nathan Forrest Ross A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2009 Copyright 2009 Nathan Forrest Ross |
