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A Berry-Esseen bound for the uniform multinomial occupancy model


 
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1. Title Title of document A Berry-Esseen bound for the uniform multinomial occupancy model
 
2. Creator Author's name, affiliation, country Jay Bartroff; University of Southern California; United States
 
2. Creator Author's name, affiliation, country Larry Goldstein; University of Southern California; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Stein’s method; size bias; coupling; urn models
 
3. Subject Subject classification 60F05; 60C05
 
4. Description Abstract

The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \geq 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that$$\sup_{z \in \mathbb{R}}\left|P\left( W_{n,m} \le z \right) -P(Z \le z)\right| \le C \frac{\sigma_{n,m}}{1+(\frac{n}{m})^3} \quad\mbox{for all $n \ge d$ and $m \ge 2$,}$$where $W_{n,m}$ and $\sigma_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variable. Asymptotically, the bound is optimal up to constants if $n$ and $m$ tend to infinity together in a way such that $n/m$ stays bounded.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSA
 
7. Date (YYYY-MM-DD) 2013-02-17
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/1983
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-1983
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
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