Shrinkage for Stabilizing the Detection of Changepoints in Covariances for High-Dimensional Data

Inhaltsverzeichnis (Table of Contents)

• Introduction
• Basics
  • Some probability theory
    • Elementary concepts
    • Stochastic processes
  • Some mathematical statistics
    • Properties of estimators
    • Test theory
  • Some linear algebra.
• Changes in univariate data
  • Change in mean
    • Testing problem.
    • Log likelihood approach
    • Asymptotic distribution
    • Appropriate variance estimators
  • Change in variance
    • Testing problem .
    • Log likelihood approach
    • Critical values.
    • General approach
• Changes in multivariate data
  • Introduction
  • The sample covariance matrix
  • Log likelihood approach
    • Preliminary work
    • LRT for full rank matrices
    • LRT for singular matrices
  • Conclusion.
• The Shrinkage Estimator
  • Introduction
  • Asymptotic framework
  • Some estimators of the covariance matrix
    • Haff estimator.
    • SteinHaff estimator
    • Minimax estimator
  • Bias-variance trade-off
  • Eigenvalue dispersion .
  • Optimal linear shrinkage
  • Analysis under general asymptotics
    • The behavior of the sample covariance matrix
    • Consistency of U* . .
  • Invertibility and condition number
  • The shrinkage estimator under Gaussian assumption
    • The RBLW estimator.
    • The OAS estimator
  • Conclusion.
• Simulation Study
  • The shrinkage estimators in comparison.
  • Basics of the simulation study
  • Change-point detection using the sample covariance matrix
    • Variance shifts
    • Covariance shifts
    • Variance-covariance shifts
  • Change-point detection using the shrinkage estimators
    • The test statistic
    • Variance shifts
    • Covariance shifts
    • Variance-covariance shifts
  • Location of the change-points
  • Summary

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis explores the detection of change-points in time series data, focusing on situations where the parameters of the data, particularly the mean and variance, might exhibit changes over time. The main objective is to develop and evaluate methods for detecting these changes, particularly in the context of multivariate data where the covariance matrix is subject to change.

• Change-point detection in time series data
• The impact of changes in parameters on statistical models and analysis
• Development and evaluation of change-point detection methods for univariate and multivariate data
• The use of shrinkage estimators to improve covariance matrix estimation in high-dimensional settings
• Simulation studies to compare the performance of different change-point detection methods

Zusammenfassung der Kapitel (Chapter Summaries)

The thesis begins by introducing the fundamental concepts of probability theory, mathematical statistics, and linear algebra, which are essential for understanding the subsequent analysis. It then delves into the detection of change-points in univariate data, discussing the CUSUM test statistic and its properties. The focus shifts to the multivariate setting in Chapter 3, where the log likelihood ratio test for changes in the covariance matrix is derived. Chapter 4 introduces shrinkage estimators, specifically the LW estimator, which aims to overcome the limitations of the sample covariance matrix in high-dimensional settings.

Schlüsselwörter (Keywords)

Change-point detection, time series analysis, multivariate data, covariance matrix, shrinkage estimators, CUSUM test, likelihood ratio test, asymptotic distribution, simulation study, high-dimensional data, estimation error, invertibility, eigenvalues, Gaussian assumption, performance comparison.