Dissertation
Inference on structural changes in high dimensional linear regression models
Washington State University
Doctor of Philosophy (PhD), Washington State University
2023
DOI:
https://doi.org/10.7273/000005134
Abstract
This dissertation is dedicated to studying the problem of constructing asymptotically valid confidence intervals for change points in high-dimensional linear models, where the number of parameters may vastly exceed the sampling period.
In Chapter 2, we develop an algorithmic estimator for a single change point and establish the optimal rate of estimation, Op(ξ−2), where ξ represents the jump size under a high dimensional scaling. The optimal result ensures the existence of limiting distributions. Asymptotic distributions are derived under both vanishing and non-vanishing regimes of jump size. In the former case, it corresponds to the argmax of a two-sided Brownian motion, while in the latter case to the argmax of a two-sided random walk, both with negative drifts. We also provide the relationship between the two distributions, which allows construction of regime (vanishing vs non-vanishing) adaptive confidence intervals.
In Chapter 3, we extend our analysis to the statistical inference for multiple change points in high-dimensional linear regression models. We develop locally refitted estimators and evaluate their convergence rates both component-wise and simultaneously. Following similar manner as in Chapter 2, we achieve an optimal rate of estimation under the component-wise scenario, which guarantees the existence of limiting distributions. While we also establish the simultaneous rate which is the sharpest available by a logarithmic factor. Component-wise and joint limiting distributions are derived under vanishing and non-vanishing regimes of jump sizes, demonstrating the relationship between distributions in the two regimes.
Lastly in Chapter 4, we introduce a novel implementation method for finding preliminary change points estimates via integer linear programming, which has not yet been explored in the current literature.
Overall, this dissertation provides a comprehensive framework for inference on single and multiple change points in high-dimensional linear models, offering novel and efficient algorithms with strong theoretical guarantees. All theoretical results are supported by Monte Carlo simulations.
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Details
- Title
- Inference on structural changes in high dimensional linear regression models
- Creators
- Hongjin Zhang
- Contributors
- Abhishek Kaul (Advisor)Venkata Krishna Jandhyala (Committee Member)Yuan Wang (Committee Member)Stergios B Fotopoulos (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 222
- Identifiers
- 99901019534501842
- Language
- English
- Resource Type
- Dissertation