Comparison of the scan and the average likelihood ratio in Gaussian mean regression

Inhaltsverzeichnis (Table of Contents)

• 1. Introduction
• 2. Comparison of the ALR and the scan statistics in the testing framework
  • 2.1. Classical theory.
  • 2.2. Introducing the ALR and the scan statistics
  • 2.3. Asymptotic null distribution of the scan and the ALR statistics
    • 2.3.1. Asymptotic distribution of the ALR statistic.
    • 2.3.2. Asymptotic distribution of the scan statistic
  • 2.4. Asymptotic power of the scan and the ALR statistics
• 3. Estimation framework: implementation and practical results
  • 3.1. Estimation procedure
  • 3.2. SQP method
    • 3.2.1. Introducing the SQP method
    • 3.2.2. SQP approach
    • 3.2.3. Pseudocode .
    • 3.2.4. Damped BFGS method.
  • 3.3. Computational issues
  • 3.4. Implementation results
• 4. Conclusion

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis focuses on the problem of detecting a signal with unknown spatial extent against a noisy background within the framework of Gaussian mean regression. It aims to provide a theoretical and practical comparison of two different statistical approaches for detecting signals: the average likelihood ratio (ALR) and the scan statistic. The study investigates the asymptotic null distribution and power of these methods.

• Detection of a signal with unknown spatial extent
• Comparison of the ALR and the scan statistics
• Asymptotic null distribution and power analysis
• Estimation framework for signal detection
• Implementation and practical results of the estimation framework

Zusammenfassung der Kapitel (Chapter Summaries)

• Chapter 1 introduces the problem of detecting a signal with an unknown spatial extent against a noisy background, setting the stage for the comparison of the ALR and the scan statistics. It defines the Gaussian mean regression model and discusses the different approaches to the problem.
• Chapter 2 provides a detailed theoretical analysis of the ALR and the scan statistics. It covers the asymptotic null distribution and power of these statistics, offering a theoretical foundation for their comparison.
• Chapter 3 delves into the practical aspects of signal estimation. It develops a numerical algorithm for solving the optimization problem related to signal estimation and presents computational results obtained through implementation.

Schlüsselwörter (Keywords)

The key terms and concepts explored in this work include Gaussian mean regression, signal detection, spatial extent, average likelihood ratio (ALR), scan statistic, asymptotic null distribution, power analysis, signal estimation, optimization problem, and computational implementation.