In mathematical statistics, detecting changes in parameters of real-life data series, known as change-point problems, is crucial. Originating in quality control during the 1950s, these problems have widespread applications today, spanning fields like economics, finance, medicine, and geology. In finance, fluctuations in asset returns can violate assumptions of constant variance, leading to inaccurate forecasts.
Chapter 2 briefly discusses the univariate case, focusing on detecting changes in mean and variance parameters over time. The Cumulative Sums (CUSUM) test statistics, derived from likelihood ratios, serve as change-point estimators. However, their asymptotic distribution complexity and slow convergence limit applicability to small sample sizes. Nevertheless, asymptotic quantiles help determine if changes have occurred.
Chapter 3 extends this analysis to the multivariate case, specifically addressing changes in covariance matrices. Estimating the covariance matrix, particularly in scenarios with many variables and few observations, poses challenges. Shrinkage estimators, like the Ledoit-Wolf (LW) estimator, offer improvements over sample covariance matrices, especially in small sample sizes. The Rao-Blackwell theorem leads to the development of the Rao-Blackwellized Ledoit-Wolf (RBLW) estimator, enhancing performance under Gaussian assumptions.
A simulation study in Chapter 5 demonstrates the effectiveness of using these shrinkage estimators in detecting change-points, resulting in improved test power and accuracy. However, due to the absence of an asymptotic distribution for the test statistics, quantiles must be obtained through simulation.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Basics
- Some probability theory
- Elementary concepts
- Stochastic processes
- Some mathematical statistics
- Properties of estimators
- Test theory
- Some linear algebra.
- Some probability theory
- 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
- Change in mean
- 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.
- Quote paper
- Mounir Zahnouni (Author), 2012, Shrinkage for Stabilizing the Detection of Changepoints in Covariances for High-Dimensional Data, Munich, GRIN Verlag, https://www.grin.com/document/1463620
