Lorenz Matz - Universal Inference: A general method for constructing statistical tests and confidence sets with finite-sample guarantees.

Universal Inference is a method for constructing valid confidence sets and tests in statistical (parametric as well as non-parametric) models, proposed in a paper of the same name by Wasserman, Ramdas & Balakrishnan (2020). It is based on a combination of the likelihood ratio approach and sample splitting. Unlike inference procedures built on classical asymptotic results or resampling techniques, the resulting confidence sets/tests have finite-sample guarantees for the confidence/significance level and do not require any regularity assumptions regarding the model. However, in models where sensible confidence sets/tests with (finite-sample or asymptotic) guarantees are available, the universal confidence sets and tests tend to be conservative in comparison, in the sense that the diameter/volume of the former will be (on average or almost surely) larger and the power of the latter will be smaller. Based on my Master's thesis, I will introduce the basic idea behind universal inference shortly and show why the resulting confidence sets and tests indeed achieve the desired confidence/significance level. Furthermore, I will present some results regarding the connection between the split and the classical likelihood ratio approach - in particular, in most statistical models of dimensions 1 and 2 commonly considered, the split set will have larger area with probability 1. Asymptotically, however, the areas of the split and the classical likelihood ratio set typically shrink at the same rate, which will be demonstrated by the example of constructing a confidence set for the unknown mean and variance in a univariate Gaussian model with both methods. Lastly, I will highlight the connection between universal inference and e-values, which have received considerable attention in the mathematical statistics literature in recent years.