A discussion of universal inference

Universal Inference is a general method for constructing confidence sets and tests in statistical models, proposed by Wasserman et al. (2020). It is based on a combination of the likelihood ratio approach and sample splitting. Unlike inference procedures built on classical asymptotic results or most methods based on resampling, the resulting confidence sets/tests have finite-sample guarantees regarding their coverage probability/significance level and do not require any regularity assumptions on the model or the null hypothesis. However, in models where sensible confidence sets and tests with (finite sample or asymptotic) guarantees are available, the universal confidence sets and tests tend to be conservative in comparison to these. This thesis adds two important points to the ongoing discussion about universal inference: On the one hand, we try to build a mathematically rigorous foundation of this method, which is still missing in the literature. Furthermore, we provide several insights regarding the connection between the universal and the classical likelihood ratio approach. In particular, we compare the split likelihood ratio confidence set which uses the maximum likelihood estimator with the classical likelihood ratio confidence set in terms of subset relations and their average volumes. We find that, in one- and two-dimensional models, the classical likelihood ratio set always constitutes a subset of this specific split likelihood ratio set, and we show that the average volume of the latter is larger than the average volume of the former in multivariate Gaussian models of any dimension. Moreover, we take the Gaussian setting as an example to illustrate why universal confidence sets and tests usually become extremely conservative in high-dimensional models, as was already noticed by other authors.