Graduation Semester and Year
Spring 2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
Dengdeng Yu
Abstract
Hypothesis testing is a crucial aspect of functional data analysis, allowing re- searchers to make inferential decisions based on samples of functional data. The inherent infinite dimensionality of functional data makes conventional hypothesis testing methods, such as Hotelling’s T2, difficult to apply due to the singularity of the sample covariance matrix. To address this issue, a common practice is to project functional data into a lower-dimensional space before conducting hypoth- esis tests. However, the choice of projection space can impact the validity and power of the tests, as the projected hypothesis problem may not be equivalent to the original problem. In this thesis, we propose a novel hypothesis testing procedure that establishes an optimal projection space where the original and projected hypothesis problems are equivalent and achieves the best power. The theoretical properties of the proposed test are systematically investigated, and a method for constructing the optimal projection space is provided. We demonstrate in this thesis that classical v functional mean testing problems, including one-sample, two-sample, and multi- sample cases, and predictor significance testing can be reduced to special cases of the proposed method. To assess the performance of our proposed test, we conduct extensive simu- lations and analyze real data. Our results show the superiority of the proposed projection test for functional linear hypotheses in the function-on-scalar regression linear model. The findings of this thesis contribute to the advancement of hypoth- esis testing in functional data analysis by addressing the challenges posed by the infinite nature of functional data and providing a novel approach to establish an optimal projection space for improved hypothesis testing performance.
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Alsaeed, Rasha Mohammed A., "FUNCTIONAL LINEAR REGRESSION: LINEAR HYPOTHESIS TESTING WITH FUNCTIONAL RESPONSE" (2023). Mathematics Dissertations. 259.
https://mavmatrix.uta.edu/math_dissertations/259
