Title: A data-based notion of quantiles on Hadamard spaces Show Chinese title

Title: 基于数据的Hadamard空间中的分位数概念

Abstract: This paper defines an alternative notion, described as data-based, of geometric quantiles on Hadamard spaces, in contrast to the existing methodology, described as parameter-based. In addition to having the same desirable properties as parameter-based quantiles, these data-based quantiles are shown to have several theoretical advantages related to large-sample properties like strong consistency and asymptotic normality, breakdown points, extreme quantiles and the gradient of the loss function. Using simulations, we explore some other advantages of the data-based framework, including simpler computation and better adherence to the shape of the distribution, before performing experiments with real diffusion tensor imaging data lying on a manifold of symmetric positive definite matrices. These experiments illustrate some of the uses of these quantiles by testing the equivalence of the generating distributions of different data sets and measuring distributional characteristics. Show Chinese abstract

Abstract: 本文定义了一种与现有方法（参数型）不同的替代概念，即以数据为基础的Hadamard空间几何分位数，与之形成对比。除了具备与参数型分位数相同的优良性质外，这些数据型分位数还表现出若干理论优势，例如大样本性质下的强一致性与渐近正态性、崩溃点、极端分位数以及损失函数的梯度。通过模拟研究，我们探讨了数据型框架的其他一些优点，包括计算更简单且更好地符合分布形状，然后使用实际扩散张量成像数据（位于对称正定矩阵流形上）进行实验。这些实验展示了这些分位数的一些用途，例如检验不同数据集生成分布的等价性及测量分布特征。