Extreme $$L^p$$ L p -quantile Kernel Regression

Quantiles are recognized tools for risk management and can be seen as minimizers of an $$L^1$$ L 1 -loss function, but do not define coherent risk measures in general. Expectiles, meanwhile, are minimizers of an $$L^2$$ L 2 -loss function and define coherent risk measures; they have started to be considered as good alternatives to quantiles in insurance and finance. Quantiles and expectiles belong to the wider family of $$L^p$$ L p -quantiles. We propose here to construct kernel estimators of extreme conditional $$L^p$$ L p -quantiles. We study their asymptotic properties in the context of conditional heavy-tailed distributions, and we show through a simulation study that taking $$p \in (1,2)$$ p ∈ ( 1 , 2 ) may allow to recover extreme conditional quantiles and expectiles accurately. Our estimators are also showcased on a real insurance data set.