Conformal prediction provides finite-sample, distribution-free coverage under exchangeability, but standard constructions may lack robustness in the presence of outliers or heavy tails. We propose a robust conformal method based on a non-conformity score defined as the half-mass radius around a point, equivalently the distance to its $(\lfloor n/2\rfloor+1)$-nearest neighbour. We show that the resulting conformal regions are marginally valid for any sample size and converge in probability to a robust population central set defined through a distance-to-a-measure functional. Under mild regularity conditions, we establish exponential concentration and tail bounds that quantify the deviation between the empirical conformal region and its population counterpart. These results provide a probabilistic justification for using robust geometric scores in conformal prediction, even for heavy-tailed or multi-modal distributions.

翻译：保形预测在可交换性假设下提供有限样本、无分布假设的覆盖保证，但标准构造在存在异常值或重尾分布时可能缺乏鲁棒性。我们提出了一种基于非一致性得分的鲁棒保形方法，该得分定义为围绕某点的半质量半径，等价于该点到其第($\lfloor n/2\rfloor+1$)个最近邻的距离。我们证明，所得保形区域对任意样本量均具有边缘有效性，并以概率收敛到通过距离与测度泛函定义的鲁棒总体中心集。在温和正则性条件下，我们建立了指数浓度和尾部界限，用以量化经验保形区域与其总体对应区域之间的偏差。这些结果为在保形预测中使用鲁棒几何得分提供了概率依据，即使对于重尾或多峰分布也成立。