Universal inference.

Area of Science:

• Statistics
• Statistical Inference

Background:

• Classical likelihood-ratio tests often fail in irregular statistical models due to intractable null distributions.
• Constructing valid tests and confidence sets is particularly challenging in mixture modeling and shape-constrained inference.

Purpose of the Study:

• To develop a general, universally applicable method for hypothesis testing and confidence set construction.
• To provide finite-sample guarantees without requiring regularity conditions in statistical inference.

Main Methods:

• Introduction of the "split likelihood-ratio test" (split LRT) statistic, a modification of the standard likelihood-ratio statistic.
• The method applies to parametric and some nonparametric models where maximum-likelihood estimators (MLEs) are computable under the null hypothesis.

Main Results:

• The split LRT offers a universally valid approach to statistical inference, simplifying complex setups.
• Extensions include using upper bounds on maximum likelihood when direct computation is difficult, handling nuisance parameters with profile likelihoods, and enabling sequential analysis for anytime-valid P-values and confidence sequences.

Conclusions:

• The proposed split LRT method provides a robust and versatile tool for statistical inference, particularly in challenging irregular models.
• The method's flexibility extends to model selection when combined with the method of sieves, offering a unified framework for various statistical problems.