This paper develops a framework for incorporating prior information into sequential multiple testing procedures while maintaining asymptotic optimality. We define a weighted log-likelihood ratio (WLLR) as an additive modification of the standard LLR and use it to construct two new sequential tests: the Weighted Gap and Weighted Gap-Intersection procedures. We prove that both procedures provide strong control of the family-wise error rate. Our main theoretical contribution is to show that these weighted procedures are asymptotically optimal; their expected stopping times achieve the theoretical lower bound as the error probabilities vanish. This first-order optimality is shown to be robust, holding in high-dimensional regimes where the number of null hypotheses grows and in settings with random weights, provided that mild, interpretable conditions on the weight distribution are met.

翻译：本文提出了一种将先验信息融入序贯多重检验程序并保持渐近最优性的框架。我们定义了加权对数似然比（WLLR）作为标准LLR的加法修正，并利用其构建了两种新的序贯检验方法：加权间隙（Weighted Gap）与加权间隙-交集（Weighted Gap-Intersection）程序。我们证明了这两种程序均能对族错误率提供强控制。我们的主要理论贡献在于证明了这些加权程序具有渐近最优性：当错误概率趋于零时，其期望停止时间达到了理论下界。这种一阶最优性被证明是稳健的，在零假设数量增长的高维情形下，以及在权重随机的设定中，只要权重分布满足温和且可解释的条件，该性质依然成立。