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Investigating Multicolor Affine Urn Models with Multiple Drawings.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Investigating Multicolor Affine Urn Models with Multiple Drawings./
Author:	Sparks, Joshua H.
Description:	1 online resource (181 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-07, Section: B.
Contained By:	Dissertations Abstracts International85-07B.
Subject:	Statistics. -
Online resource:	click for full text (PQDT)
ISBN:	9798381378054

Investigating Multicolor Affine Urn Models with Multiple Drawings.
Sparks, Joshua H.

Investigating Multicolor Affine Urn Models with Multiple Drawings. - 1 online resource (181 pages)

Source: Dissertations Abstracts International, Volume: 85-07, Section: B.

Thesis (Ph.D.)--The George Washington University, 2024.

Includes bibliographical references

The goal of this research is to analyze multicolor affine urn structures that evolve through multiple drawings at each sampling stage. The jump from one to multiple balls per sample, especially with more than two colors, can be daunting due to the replacement matrix no longer being square in nature. However, this class of urns finds a gateway to rich analysis thanks to its inherent "core matrix", which dictates the urn's progression through its linear replacement criteria. Chapter 1 provides a brief summary and history of urn models, including those that sample multiple balls at a time. Chapter 2 provides a preliminary discussion of notation along with definitions of matrix properties, irreducible and linear urns, and martingales that will be necessary for our analysis and methodology. In Chapter 3, we produce a detailed analysis of affine urns as well as properties for growing irreducible urns based on what we call a core index of our core matrix, classified as one of either small, critical, or large.We examine the case when the core matrix is scalar in Chapter 4, allowing us to simplify complex recursive structures in the covariance matrix and provide closed-form moments and simulation results for affine urns which are not irreducible. In Chapter 5, we explore urn structures that possess equitable status, allowing us to generalize a wide class of affine urns that commute with their transpose and thus have principal left and right eigenvectors scalable to the k-dimensional vector of ones. The final special case summarizes Sparks, Balaji, and Mahmoud (2022) and stems from a scenario where the urn is 1-balanced and generalizes the well-studied area of recursive trees, formulating what we call hyperrecursive trees. Discussion of this structure-including its local and global profiles, convergence in distribution, relationship to affine urns, and simulation work-is explored in Chapter 6. Chapter 7 discusses potential future projects that extend the results provided here.Portions of this research have been accepted for publication and in refereed academic conferences. Much of Chapter 6 can be found in Sparks, Balaji, and Mahmoud (2022), while its advances and related work from Chapter 3 have been presented at the International Workshop on Applied Probability. The general study of Chapter 3, as well as the advances for Chapter 5 and its related work in Chapter 4, are being prepared for journal submissions.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798381378054Subjects--Topical Terms:

556824
Statistics.
Subjects--Index Terms:

Hyperrecursive treeIndex Terms--Genre/Form:

554714
Electronic books.

Investigating Multicolor Affine Urn Models with Multiple Drawings.
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520 $a The goal of this research is to analyze multicolor affine urn structures that evolve through multiple drawings at each sampling stage. The jump from one to multiple balls per sample, especially with more than two colors, can be daunting due to the replacement matrix no longer being square in nature. However, this class of urns finds a gateway to rich analysis thanks to its inherent "core matrix", which dictates the urn's progression through its linear replacement criteria. Chapter 1 provides a brief summary and history of urn models, including those that sample multiple balls at a time. Chapter 2 provides a preliminary discussion of notation along with definitions of matrix properties, irreducible and linear urns, and martingales that will be necessary for our analysis and methodology. In Chapter 3, we produce a detailed analysis of affine urns as well as properties for growing irreducible urns based on what we call a core index of our core matrix, classified as one of either small, critical, or large.We examine the case when the core matrix is scalar in Chapter 4, allowing us to simplify complex recursive structures in the covariance matrix and provide closed-form moments and simulation results for affine urns which are not irreducible. In Chapter 5, we explore urn structures that possess equitable status, allowing us to generalize a wide class of affine urns that commute with their transpose and thus have principal left and right eigenvectors scalable to the k-dimensional vector of ones. The final special case summarizes Sparks, Balaji, and Mahmoud (2022) and stems from a scenario where the urn is 1-balanced and generalizes the well-studied area of recursive trees, formulating what we call hyperrecursive trees. Discussion of this structure-including its local and global profiles, convergence in distribution, relationship to affine urns, and simulation work-is explored in Chapter 6. Chapter 7 discusses potential future projects that extend the results provided here.Portions of this research have been accepted for publication and in refereed academic conferences. Much of Chapter 6 can be found in Sparks, Balaji, and Mahmoud (2022), while its advances and related work from Chapter 3 have been presented at the International Workshop on Applied Probability. The general study of Chapter 3, as well as the advances for Chapter 5 and its related work in Chapter 4, are being prepared for journal submissions.
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