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重要实践意义的问题。错误率控制对这一逻辑至关重要，但在复杂的非参数设定下，这类保证难以实现，尤其是在决定数据收集过程的停止规则不可知时尤为困难。本文提出一种创新的e-过程构建方法，该方法利用了被称为预测递归（PR）的算法，该算法旨在快速递归地拟合非参数混合模型。由此产生的PRe-过程能够在所有停止规则下实现随时有效的推断，并且被证明具有高效性——其在备择假设下实现了相对于PR所拟合混合模型的最大增长率。在对对数凹密度进行检验的特例中，与近期提出的一种随时有效检验方法相比，PRe-过程检验在计算上更简单、速度更快、稳定性更高，且效率毫不逊色。