This paper addresses the challenge of improving finite sample performance in Ranking and Selection by developing a Bahadur-Rao type expansion for the Probability of Correct Selection (PCS). While traditional large deviations approximations captures PCS behavior in the asymptotic regime, they can lack precision in finite sample settings. Our approach enhances PCS approximation under limited simulation budgets, providing more accurate characterization of optimal sampling ratios and optimality conditions dependent of budgets. Algorithmically, we propose a novel finite budget allocation (FCBA) policy, which sequentially estimates the optimality conditions and accordingly balances the sampling ratios. We illustrate numerically on toy examples that our FCBA policy achieves superior PCS performance compared to tested traditional methods. As an extension, we note that the non-monotonic PCS behavior described in the literature for low-confidence scenarios can be attributed to the negligence of simultaneous incorrect binary comparisons in PCS approximations. We provide a refined expansion and a tailored allocation strategy to handle low-confidence scenarios, addressing the non-monotonicity issue.

翻译：本文针对排序与选择问题中有限样本性能提升的挑战，通过建立正确选择概率的Bahadur-Rao型级数展开展开展开进行研究。传统的大偏差近似方法虽能刻画渐近区域内的PCS行为，但在有限样本场景下往往精度不足。我们的方法在有限仿真预算下改进了PCS近似，为依赖于预算的最优采样比例与最优性条件提供了更精确的表征。在算法层面，我们提出了一种新颖的有限预算分配策略，该策略通过序贯估计最优性条件并据此平衡采样比例。通过数值算例验证，我们的FCBA策略相较于传统测试方法获得了更优的PCS性能。作为拓展，我们指出文献中低置信度场景下描述的非单调PCS行为可归因于PCS近似中同时存在的错误二元比较被忽略。为此我们提出了改进的级数展开与定制化分配策略以处理低置信度场景，从而解决了非单调性问题。