Multivariate convex risk statistics with scenario analysis

Abstract

In this article, we introduce three new classes of multivariate risk statistics, which can be considered as data-based versions of multivariate risk measures. These new classes are multivariate convex risk statistics, multivariate comonotonic convex risk statistics and multivariate empirical-law-invariant convex risk statistics, respectively. Representation results are provided. The arguments of proofs are mainly developed by ourselves. It turns out that all the relevant existing results in the literature are special cases of those obtained in this article.

Keywords:

• (Multivariate) risk measures
• multivariate convex risk statistics
• comonotonicity
• empirical law invariance

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 91B30
• 91B32
• 91B70

Acknowledgments

The authors are very grateful to the Editor-in-Chief Professor Narayanaswamy Balakrishnan and the anonymous referees for their constructive comments and suggestions which led to the present greatly improved version of the manuscript.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (nos: 11601463, 11771343, 11371284).