Regularized geometric quantiles and universal linear distribution functionals

TL;DR

This paper introduces a regularized version of geometric quantiles and distribution functions that improve numerical stability and broad applicability in multivariate statistical analysis, without requiring moment conditions.

Contribution

It proposes a novel regularization approach for geometric quantiles and distribution functions, ensuring stability and broad applicability, and characterizes all linear, translation- and orthogonal-equivariant distribution functionals.

Findings

Regularized geometric quantiles are close to classical ones and retain key properties.

The regularized functions enable asymptotic results without moment conditions.

Any linear, translation- and orthogonal-equivariant distribution functional coincides with the proposed regularized functions.

Abstract

Geometric quantiles are popular location functionals to build rank-based statistical procedures in multivariate settings. They are obtained through the minimization of a non-smooth convex objective function. As a result, the singularity of the directional derivatives leads to numerical instabilities and poor sample properties as well as surprising `phase transitions' from empirical to population distributions. To solve these issues, we introduce a regularized version of geometric distribution functions and quantiles that are provably close to the usual geometric concepts and share their qualitative properties, both in the empirical and continuous case, while allowing for a much broader applicability of asymptotic results without any moment condition. We also show that any linear assignment of probability measures (such as the univariate distribution function), that is also translation- and orthogonal-equivariant, necessarily coincides with one of our regularized geometric distribution functions.