High-dimensional inference and confidence sets of models

In high-dimensional regression problems, a key aim is to identify a sparse model that fits the data well. This may be used to form accurate predictions or to gain subject-matter understanding. When strong dependence is present among covariates, it is common for many models to fit the data equally well. Whilst it is sufficient to report a single model for prediction, when the goal is to gain subject-matter understanding, Cox & Battey (2017) argue that a confidence set of models - a set consisting of all models of appropriate fit - should be reported and propose a method to achieve this aim. This thesis provides a theoretical elucidation of this approach, and based on the results, explores further ideas in high-dimensional data analysis.