Geometric quantiles are location parameters which extend classical univariate quantiles to normed spaces (possibly infinite-dimensional) and which include the geometric median as a special case. The infinite-dimensional setting is highly relevant in the modeling and analysis of functional data, as well as for kernel methods. We begin by providing new results on the existence and uniqueness of geometric quantiles. Estimation is then performed with an approximate M-estimator and we investigate its large-sample properties in infinite dimension. When the population quantile is not uniquely defined, we leverage the theory of variational convergence to obtain asymptotic statements on subsequences in the weak topology. When there is a unique population quantile, we show that the estimator is consistent in the norm topology for a wide range of Banach spaces including every separable uniformly convex space. In separable Hilbert spaces, we establish weak Bahadur-Kiefer representations of the estimator, from which $\sqrt n$-asymptotic normality follows.

翻译：几何分位数是一类位置参数，它将经典单变量分位数推广到赋范空间（可能为无限维），并包含几何中位数作为特例。无限维设定在函数型数据的建模与分析以及核方法中具有高度相关性。我们首先给出关于几何分位数存在性与唯一性的新结果。随后利用近似M估计量进行估计，并研究其在无限维中的大样本性质。当总体分位数非唯一时，我们借助变分收敛理论在弱拓扑下获得子序列的渐近性结论；当存在唯一总体分位数时，我们证明该估计量在包括所有可分一致凸空间在内的广泛Banach空间类中依范数拓扑具有相合性。在可分Hilbert空间中，我们建立了估计量的弱Bahadur-Kiefer表示，由此可得$\sqrt n$渐近正态性。