Flexible modelling and expectile regression with applications in risk measures

In statistics a core question is how a variable of interest Y is influenced by a vector of explanatory factors X. There are several ways to quantify such influences. For example, by studying how Y is influenced on average by X (i.e the conditional mean E(Y|X) or shortly mean regression), or by looking at a conditional median (and generally conditional quantiles). In recent years there is a lot of interest in expectile regression, which can be viewed as an asymmetric generalization of mean regression.

Expectiles have become very popular in risk management. When Y describes a financial loss, a large claim in insurance, the wind speed in hurricanes, ..., one wants to measure the risk that goes with this. Appropriate risk measures that satisfy some essential properties, and are called coherent risk measures. For example, a risk should be subadditive: the capital requirement (in a financial context) for two risks combined should not be greater than for the risks treated separately. An expectile is a coherent risk measure.

For many applications there is no prior knowledge on how an explanatory factor might influence the regression expectile. Therefore one should not impose too restrictive modelling assumptions (such as linearity), but go for flexible enough modelling. In a first part we provide solid theoretical contributions to statistical inference for expectile regression in flexible settings. In a second part we study expectile-based risk measures as well aggregated risks.