  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.
