Detecting Weak Distribution Shifts via Displacement Interpolation

Abstract

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 datasets 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.

Keywords:

• Displacement interpolation
• Hypothesis testing
• Optimal transport
• Phase transition
• Weak distribution shifts

Acknowledgments

TL and YH thank the editors and anonymous referees for the constructive feedback on strengthening the empirical results.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is possible to generalize the aforementioned concepts—displacement interpolation, optimal transport, and the Wasserstein-p distance—to ${\mathbb{R}}^d$ with d > 1; see (Villani 2003).

2 Data are publicly available at https://tracktherecovery.org/.

3 They have the largest number of observations among all nine disciplines.

Additional information

Funding

Liang acknowledges the generous support from the NSF Career Award (DMS-2042473), and the William Ladany Faculty Fellowship at the University of Chicago Booth School of Business.