On a general class of functionals: Statistical inference and application to risk measures

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 a subclass of distortion risks, as well as new concepts. 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 members of the class in one single effort. 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.