Universal inference

doi:10.1073/pnas.1922664117

Universal inference

2020-07-06

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Sivaraman Balakrishnan

DOI：10.1073/pnas.1922664117

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Abstract

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We propose a general method for constructing confidence sets and hypothesis tests that have finite-sample guarantees without regularity conditions. We refer to such procedures as univer-sal. The method is very simple and is based on a modified version of the usual likelihood-ratio statistic that we call the split likelihood-ratio test (split LRT) statistic. The (limiting) null distri-bution of the classical likelihood-ratio statistic is often intractable when used to test composite null hypotheses in irregular statis-tical models. Our method is especially appealing for statistical inference in these complex setups. The method we suggest works for any parametric model and also for some nonparametric mod-els, as long as computing a maximum-likelihood estimator (MLE) is feasible under the null. Canonical examples arise in mixture modeling and shape-constrained inference, for which construct-ing tests and confidence sets has been notoriously difficult. We also develop various extensions of our basic methods. We show that in settings when computing the MLE is hard, for the pur-pose of constructing valid tests and intervals, it is sufficient to upper bound the maximum likelihood. We investigate some con-ditions under which our methods yield valid inferences under model misspecification. Further, the split LRT can be used with profile likelihoods to deal with nuisance parameters, and it can also be run sequentially to yield anytime-valid P values and con-fidence sequences. Finally, when combined with the method of sieves, it can be used to perform model selection with nested model classes.

Keywords:

likelihood

testing

irregular models

confidence sequence

Universal inference

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