  We define extrapolation as any type of statistical inference on a conditional
function (e.g., a conditional expectation or conditional quantile) evaluated
outside of the support of the conditioning variable. This type of extrapolation
occurs in many data analysis applications and can invalidate the resulting
conclusions if not taken into account. While extrapolating is straightforward
in parametric models, it becomes challenging in nonparametric models. In this
work, we extend the nonparametric statistical model to explicitly allow for
extrapolation and introduce a class of extrapolation assumptions that can be
combined with existing inference techniques to draw extrapolation-aware
conclusions. The proposed class of extrapolation assumptions stipulate that the
conditional function attains its minimal and maximal directional derivative, in
each direction, within the observed support. We illustrate how the framework
applies to several statistical applications including prediction and
uncertainty quantification. We furthermore propose a consistent estimation
procedure that can be used to adjust existing nonparametric estimates to
account for extrapolation by providing lower and upper extrapolation bounds.
The procedure is empirically evaluated on both simulated and real-world data.
