  Statistical learning under distribution shift is challenging when neither
prior knowledge nor fully accessible data from the target distribution is
available. Distributionally robust learning (DRL) aims to control the
worst-case statistical performance within an uncertainty set of candidate
distributions, but how to properly specify the set remains challenging. To
enable distributional robustness without being overly conservative, in this
paper, we propose a shape-constrained approach to DRL, which incorporates prior
information about the way in which the unknown target distribution differs from
its estimate. More specifically, we assume the unknown density ratio between
the target distribution and its estimate is isotonic with respect to some
partial order. At the population level, we provide a solution to the
shape-constrained optimization problem that does not involve the isotonic
constraint. At the sample level, we provide consistency results for an
empirical estimator of the target in a range of different settings. Empirical
studies on both synthetic and real data examples demonstrate the improved
accuracy of the proposed shape-constrained approach.
