# Multivariate Geometric Expectiles

## TL;DR

This paper introduces multivariate geometric expectiles as a new class of risk measures for d-dimensional distributions, demonstrating their theoretical properties and practical applications in multivariate risk assessment.

## Contribution

It presents the first formulation of geometric expectiles for multivariate distributions, including their theoretical properties and consistency of the sample estimator.

## Key findings

- Geometric expectiles are unique solutions to a convex risk minimization problem.
- They are well-behaved under common data transformations.
- Sample geometric expectiles are consistent estimators.

## Abstract

A generalization of expectiles for d-dimensional multivariate distribution functions is introduced. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. They are well behaved under common data transformations and the corresponding sample version is shown to be a consistent estimator. We exemplify their usage as risk measures in a number of multivariate settings, highlighting the influence of varying margins and dependence structures.