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

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

A. Proofs

A.1. On the boundedness of the penalized scan statistic

A.2. Proof of Theorem 2.5

A.2.1. Asymptotic distribution of Aεn

A.2.2. Asymptotic distribution of Bεn

A.2.3. Asymptotic distribution of ALRn

A.3. Proof of Theorem 2.9

B. Auxiliary proofs

Objectives and Topics

The primary objective of this thesis is to compare the scan statistic and the average likelihood ratio (ALR) statistic within the framework of Gaussian mean regression, specifically regarding their performance in detecting signals of unknown spatial extent. The work further aims to develop a numerical procedure for constructing estimators that effectively denoise data while maintaining residuals consistent with white Gaussian noise.

• Theoretical analysis of scan and ALR statistics for signal detection.
• Asymptotic null distributions and power analysis of the discussed statistics.
• Implementation of optimization algorithms (SQP and BFGS) for signal estimation.
• Evaluation of estimator quality through simulation studies under varying noise levels.

Excerpt from the Book

Summary of Chapters

1. Introduction: Outlines the problem of detecting signals with unknown spatial extent in Gaussian mean regression and introduces the scan and ALR statistics.

2. Comparison of the ALR and the scan statistics in the testing framework: Examines theoretical properties, asymptotic null distributions, and detection power of the scan and ALR statistics.

3. Estimation framework: implementation and practical results: Develops numerical optimization algorithms for signal reconstruction and presents simulation results evaluating estimator performance.

4. Conclusion: Summarizes the thesis findings and suggests potential areas for further research, such as extending the methods to two-dimensional signals.

Keywords

Gaussian mean regression, signal detection, scan statistic, average likelihood ratio, ALR, penalized scan, hypothesis testing, asymptotic distribution, signal estimation, data denoising, constrained optimization, SQP method, BFGS method, multiscale problem, non-parametric statistics.

Frequently Asked Questions

What is the core focus of this research?

This thesis focuses on the statistical problem of detecting signals with unknown spatial extents within a Gaussian mean regression model and constructing accurate estimators for those signals.

What are the primary test statistics analyzed?

The research compares the scan statistic and the average likelihood ratio (ALR) statistic.

What is the main objective of the proposed signal estimation?

The goal is to find an estimator such that the resulting residuals resemble white Gaussian noise, effectively denoising the observed data.

Which mathematical methods are used for estimation?

The thesis utilizes sequential quadratic programming (SQP) and the damped BFGS method to solve the constrained optimization problems associated with signal estimation.

What is the primary difference in performance between the scan and ALR statistics?

Empirical and theoretical evidence suggests that the scan statistic is optimal for detecting signals with the smallest spatial extents, whereas the ALR statistic performs better on larger scales.

What is the significance of the "penalized scan" statistic?

The penalized scan is a modification of the standard scan statistic designed to overcome the issue of unboundedness, providing a more stable performance across different signal scales.

How is the "simplicity" of an estimator measured?

The thesis employs a simplicity functional based on the squared increments of the function, which encourages smoothness in the estimated signal.

Why are standard exponential functions problematic for the ALR statistic?

Evaluating the ALR for signals with non-zero intensity can lead to excessively large values that exceed the limits of standard floating-point arithmetic, necessitating the use of modified functions in implementation.