Distinguishing two classes of candidate models is a fundamental and practically important problem in statistical inference. Error rate control is crucial to the logic but, in complex nonparametric settings, such guarantees can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. In this paper we develop a novel e-value construction that leverages the so-called predictive recursion (PR) algorithm designed to recursively fit nonparametric mixture models. The resulting PRe-value affords anytime valid inference uniformly over stopping rules and is shown to be efficient in the sense that it achieves the maximal growth rate under the alternative relative to the mixture model being fit by PR. In the special case of testing the density for log-concavity, the PRe-value test is shown empirically to be significantly more efficient than a recently proposed anytime valid test based on universal inference.

翻译：区分两类候选模型是统计推断中一个基础且具有重要实践意义的问题。错误率控制对逻辑推理至关重要，但在复杂的非参数设定中，尤其是在决定数据收集过程的停止规则不可知的情况下，此类保证难以实现。本文提出了一种新颖的e值构造方法，该方法利用了被称为预测递归（PR）的算法，该算法旨在递归地拟合非参数混合模型。由此产生的PRe值可在所有停止规则下实现统一有效的任意有效推断，并且该方法被证明是高效的——相对于PR所拟合的混合模型，它能在备择假设下实现最大增长率。在针对对数凹性密度检验的特例中，实证结果表明PRe值检验的效率显著高于近期基于通用推断提出的任意有效检验方法。