  Detecting weak, systematic distribution shifts and quantitatively modeling
individual, heterogeneous responses to policies or incentives have found
increasing empirical applications in social and economic sciences. Given two
probability distributions $P$ (null) and $Q$ (alternative), we study the
problem of detecting weak distribution shift deviating from the null $P$ toward
the alternative $Q$, where the level of deviation vanishes as a function of
$n$, the sample size. We propose a model for weak distribution shifts via
displacement interpolation between $P$ and $Q$, drawing from the optimal
transport theory. We study a hypothesis testing procedure based on the
Wasserstein distance, derive sharp conditions under which detection is
possible, and provide the exact characterization of the asymptotic Type I and
Type II errors at the detection boundary using empirical processes. We
demonstrate how the proposed testing procedure works in modeling and detecting
weak distribution shifts in real data sets using two empirical examples:
distribution shifts in consumer spending after COVID-19, and heterogeneity in
the published p-values of statistical tests in journals across different
disciplines.
