Department of Statistics and Data Science, Carnegie Mellon University, Pittsburgh, PA 15213;
Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15213.
Proc Natl Acad Sci U S A. 2020 Jul 21;117(29):16880-16890. doi: 10.1073/pnas.1922664117. Epub 2020 Jul 6.
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 "universal." 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 distribution of the classical likelihood-ratio statistic is often intractable when used to test composite null hypotheses in irregular statistical 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 models, 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 constructing 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 purpose of constructing valid tests and intervals, it is sufficient to upper bound the maximum likelihood. We investigate some conditions 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 confidence sequences. Finally, when combined with the method of sieves, it can be used to perform model selection with nested model classes.
我们提出了一种通用的方法，用于构建具有有限样本保证而无需正则条件的置信集和假设检验。我们将这种方法称为“通用”。该方法非常简单，基于我们称之为“分裂似然比检验”（split LRT）统计量的常用似然比统计量的修改版本。当用于检验不规则统计模型中的复合零假设时，经典似然比统计量的（极限）零分布通常难以处理。我们的方法特别适用于这些复杂设置中的统计推断。我们建议的方法适用于任何参数模型，也适用于一些非参数模型，只要在零假设下计算最大似然估计（MLE）是可行的。典型的例子出现在混合建模和形状约束推理中，对于这些模型，构建检验和置信集一直是非常困难的。我们还开发了我们基本方法的各种扩展。我们表明，在计算 MLE 困难的情况下，为了构建有效的检验和区间，只需对最大似然进行上界即可。我们研究了在模型误设的情况下我们的方法产生有效推断的一些条件。此外，split LRT 可以与轮廓似然一起用于处理烦扰参数，并且可以顺序运行以生成任何时候有效的 P 值和置信序列。最后，当与筛子方法结合使用时，它可用于执行嵌套模型类的模型选择。
