Generalized extremiles and risk measures of distorted random variables

TL;DR

This paper introduces a unified framework for risk measures based on extremiles and distortion functions, develops estimators with proven asymptotic properties, and demonstrates broad applicability including real-world disaster data analysis.

Contribution

It formulates a general class of risk functionals encompassing extremiles, expectiles, and distortion risks, and provides estimators with asymptotic guarantees for diverse applications.

Findings

Establishes asymptotic consistency and normality of estimators

Demonstrates broad applicability through various examples

Applies methods to analyze natural disaster data

Abstract

Quantiles, expectiles and extremiles can be seen as concepts defined via an optimization problem, where this optimization problem is driven by two important ingredients: the loss function as well as a distributional weight function. This leads to the formulation of a general class of functionals that contains next to the above concepts many interesting quantities, including also a subclass of distortion risks. The focus of the paper is on developing estimators for such functionals and to establish asymptotic consistency and asymptotic normality of these estimators. The advantage of the general framework is that it allows application to a very broad range of concepts, providing as such estimation tools and tools for statistical inference (for example for construction of confidence intervals) for all involved concepts. After developing the theory for the general functional we apply it to various settings, illustrating the broad applicability. In a real data example the developed tools are used in an analysis of natural disasters.