Deep quantile and deep composite triplet regression

A main difficulty in actuarial claim size modeling is that covariates may have different effects on the body of the conditional distribution and on its tail. To cope with this problem, we introduce a deep composite regression model whose splicing point is given in terms of a quantile of the conditional claim size distribution (rather than a constant). This allows us to simultaneously fit different regression models in the two different parts of the conditional distribution function. To facilitate M-estimation for such models, we introduce and characterize the class of strictly consistent scoring functions for the triplet consisting of a quantile, as well as the lower and upper expected shortfall beyond that quantile. In a second step, this elicitability result is applied to fit deep neural network regression models. We demonstrate the applicability of our approach and its superiority over classical approaches on a real data set from accident insurance.