We propose the Modified Mahalanobis Distance Conformal Prediction (MMDCP), a unified framework for multi-class classification and outlier detection under label shift, where the training and test distributions may differ. In such settings, many existing methods construct nonconformity scores based on empirical cumulative or density functions combined with data-splitting strategies. However, these approaches are often computationally expensive due to their heavy reliance on resampling procedures and tend to produce overly conservative prediction sets with unstable coverage, especially in small samples. To address these challenges, MMDCP combines class-specific distance measures with full conformal prediction to construct a score function, thereby producing adaptive prediction sets that effectively capture both inlier and outlier structures. Under mild regularity conditions, we establish convergence rates for the resulting sets and provide the first theoretical characterization of the gap between oracle and empirical conformal $p$-values, which ensures valid coverage and effective control of the class-wise false discovery rate (CW-FDR). We further introduce the Summarized Class-Wise FDR (SCW-FDR), a novel global error metric aggregating false discoveries across classes, and show that it can be effectively controlled within the MMDCP framework. Extensive simulations and two real-data applications support our theoretical findings and demonstrate the advantages of the proposed method.

翻译：本文提出改进的马氏距离保形预测（MMDCP），这是一个在标签偏移条件下（训练与测试分布可能不同）用于多类分类和离群值检测的统一框架。在此类场景中，现有方法多基于经验累积函数或密度函数结合数据分割策略构建非保形分数。然而，这些方法因严重依赖重采样过程而计算成本高昂，且倾向于产生覆盖不稳定、过于保守的预测集，尤其在小样本中更为明显。为应对这些挑战，MMDCP将类别特定距离度量与完全保形预测相结合以构建评分函数，从而生成能有效捕捉内围值与离群值结构的自适应预测集。在温和的正则性条件下，我们建立了所得集合的收敛速率，并首次从理论上刻画了理想保形p值与经验保形p值之间的差距，这确保了有效覆盖度与类别错误发现率（CW-FDR）的可靠控制。我们进一步提出汇总类别错误发现率（SCW-FDR）——一种聚合跨类别错误发现的新型全局误差度量，并证明其在MMDCP框架内可实现有效控制。大量模拟实验与两个实际数据应用验证了理论结论，并证明了所提方法的优势。