Estimation and Inference of Change Points in High-Dimensional Time Series under Multi-Source Signal Structures.

Dipesh Baral

This dissertation develops new methodologies for estimation and inference of change points in high-dimensional time series under multi-source signal structures. The proposed framework integrates heterogeneous sources of structural information—arising from changes in regression parameters, distributional shifts in covariates, and temporal variation across multiple time scales—within a unified inferential approach. The methods are supported by rigorous theoretical guarantees and are evaluated through simulation studies and real-data applications. Chapter 2 studies change-point estimation in high-dimensional linear regression models where structural breaks may manifest through multiple signal sources. A novel estimator is proposed based on a loss function that jointly captures changes in regression parameters and covariate distributions. Unlike existing methodologies that only use single signal source, the proposed method systematically combines complementary sources to improve estimation accuracy. The resulting estimator achieves the sharp convergence rate $ O_p(\xi^{-2}) $, enabling valid asymptotic inference for the change-point location. Here, $ \xi $ represents a weighted contraharmonic mean of the jump sizes associated with the different signal sources. The limiting distribution is characterized by the argmax of a two-sided Brownian motion with negative drift when $ \xi \to 0 $, and by the argmax of a two-sided random walk with negative drift when $ \xi $ remains bounded away from zero. Monte Carlo experiments and a real-data application corroborate the theoretical findings. Chapter 3 extends the analysis to high-dimensional time series exhibiting complex temporal dynamics driven by multiple underlying signal sources, including variations across distinct time scales. A unified framework based on an $ L_2 $ loss function is developed to capture and aggregate these signals, allowing for coherent estimation of structural changes in settings where traditional single-source approaches are inadequate. The proposed estimator attains a sharp convergence rate governed by the effective signal strength, thereby facilitating asymptotic inference for the change-point location. The theoretical results are supported by simulation studies and further validated through an empirical analysis of climate data. Overall, this dissertation advances the theory and methodology of change-point analysis in high-dimensional settings by explicitly accounting for multi-source signal structures. By integrating diverse forms of structural information within a common framework, the proposed methods enhance both statistical efficiency and inferential validity. These contributions expand the toolkit for analyzing complex temporal data and open new directions for research in high-dimensional time series and structural change analysis.

Estimation and Inference of Change Points in High-Dimensional Time Series under Multi-Source Signal Structures.

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