LINEAR HYPOTHESIS TESTING FOR HIGH DIMENSIONAL GENERALIZED LINEAR MODELS.

Area of Science:

• Statistics
• High-Dimensional Data Analysis
• Statistical Modeling

Background:

• Generalized linear models (GLMs) are widely used but face challenges with high-dimensional data.
• Testing linear hypotheses in such settings requires specialized methods to handle numerous parameters.
• Existing regularization techniques may not be optimal for hypothesis testing in high dimensions.

Purpose of the Study:

• To develop and evaluate novel methods for testing linear hypotheses in high-dimensional GLMs.
• To introduce a constrained partial regularization approach and associated algorithms.
• To propose and analyze partial penalized likelihood ratio, score, and Wald tests.

Main Methods:

• Constrained partial regularization method for high-dimensional GLMs.
• Algorithm for regularization problems with folded-concave penalty functions and linear constraints.
• Development of partial penalized likelihood ratio, score, and Wald test statistics.

Main Results:

• The proposed partial penalized tests demonstrate chi-squared limiting null distributions.
• Under local alternatives, these tests asymptotically follow non-central chi-squared distributions.
• The statistical properties of the constrained partial regularization method are theoretically studied.

Conclusions:

• The developed partial penalized tests are effective for hypothesis testing in high-dimensional GLMs.
• The proposed methods provide a robust framework for analyzing complex statistical models.
• Simulation studies and real data analysis validate the practical utility of the new testing procedures.