Statistical properties of approximate geometric quantiles in infinite-dimensional Banach spaces

TL;DR

This paper investigates the statistical properties of approximate geometric quantiles in infinite-dimensional Banach spaces, establishing existence, uniqueness, consistency, and asymptotic normality results, including a novel CLT in Hilbert spaces.

Contribution

It provides new theoretical results on the existence, uniqueness, and asymptotic behavior of geometric quantiles in infinite-dimensional spaces, including the first CLT in Hilbert spaces.

Findings

Proved existence and uniqueness of geometric quantiles in Banach spaces.

Established consistency and asymptotic normality of estimators in Hilbert spaces.

Derived a new CLT for geometric medians in Hilbert spaces.

Abstract

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, under minimal assumptions, 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. As a consequence, we obtain the first central limit theorem valid in a generic Hilbert space and under minimal assumptions that exactly match those of the finite-dimensional case. Our consistency and asymptotic normality results significantly improve the state of the art, even for exact geometric medians in Hilbert spaces.