Maozai Tian, Keming Yu and Jiangfeng Wang’s contribution to the Discussion of ‘Safe testing’ by Grünwald, De Heide, and Koolen

原文摘要

We want to congratulate the authors on the publication of ‘Safe Testing’ on JRSS-B. It deepens our understanding of the statistical hypothesis test that is one particular method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. The e-variables and safe tests offer several advantages. The authors develop the theory of hypothesis testing based on the e-value that behave excellently under optional continuation and are enhanced with interpretability and flexibility. It is particularly noteworthy that the authors pay careful attention to the issues involved in Type I error guarantees. They argue that e-variables can be based on Bayesian prior knowledge on earlier data but also on minimax performance considerations, in all cases preserving frequentist Type I error guarantees. Our main focus is on two issues: First, for Type I error as shown in this article, the minimax results based on e-variables for composite hypothesis, i.e. the parameter’s value is given by a set of numbers, are too pessimistic. The minimax approach is conservative because it only guarantees the best possible performance in the ‘worst case’ situation. Second, for the GRO criterion that is growth rate optimality (GRO) as an analogue of power in an optional continuation context: The asymptotic rate results are almost uninformative and cannot be used for practical choice of the tuning parameters in finite sample studies or for comparing different procedures. The authors thank the RSS Journals Manager Judith Shorten for her kind invitation to make presentation with discussion of the article. The work was the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (22XNL016).

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