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-process construction that leverages the so-called predictive recursion (PR) algorithm designed to rapidly and recursively fit nonparametric mixture models. The resulting PRe-process 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 for a log-concave density, the PRe-process test is computationally simpler and faster, more stable, and no less efficient compared to a recently proposed anytime valid test.

翻译：区分两类候选模型是统计推断中一个基础且具有实际重要性的问题。错误率控制对其逻辑至关重要，但在复杂的非参数设定中，这种保证难以实现，尤其是在决定数据收集过程的停止规则不可得时。本文提出了一种新颖的e过程构建方法，该方法利用了所谓预测递归（PR）算法，该算法设计用于快速递归拟合非参数混合模型。由此产生的PRe过程在停止规则上具有一致的任意时间有效性，并且被证明是高效的，即它在备择假设下相对于PR拟合的混合模型达到了最大增长率。在对数凹密度检验的特殊情况下，与最近提出的任意时间有效检验相比，PRe过程检验在计算上更简单、更快速、更稳定，且效率不逊色。