arXiv 6 Jul 2025 · Statistics — Methodology
arXiv:2507.04560 · PDF · DOI · OpenAlex · Extracted main text
Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regression, the method estimates a mean path on either side of a candidate cutoff. This extends existing $k$-sample tests to a non-parametric regression setting with metric-space valued outcomes. I establish the asymptotic distribution of the test and its consistency against contiguous alternatives. For this, I derive a central limit theorem for the local estimator of the conditional Fr\'echet variance and a consistent estimator of its asymptotic variance. Simulations confirm nominal size control and robust power in finite samples. Two empirical illustrations demonstrate the method's ability to reveal discontinuities missed by scalar-based tests. I find sharp changes in (i) work-from-home compositions at an income threshold for non-compete enforceability and (ii) national input-output networks following the loss of preferential U.S. trade access. These findings show the value of analyzing regression outcomes in their native metric spaces.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Dubey, P. and Müller, H.-G (2019) Fréchet analysis of variance for random objects | 0.956 | 8 | 5 | 88% |
| 2 | Kurisu, D., Zhou, Y., Otsu, T., and Müller, H.-G (2025) Regression discontinuity designs for functional data and random objects in geodesic spaces | 0.941 | 6 | 3 | 83% |
| 3 | Van Dijcke, D (2025) Regression discontinuity design with distribution-valued outcomes | 0.941 | 6 | 3 | 83% |
| 4 | Petersen, A. and Müller, H.-G (2019) Fréchet regression for random objects with euclidean predictors | 0.863 | 14 | 5 | 64% |
| 5 | Zhou, Y. and Müller, H.-G (2022) Network regression with graph laplacians | 0.811 | 4 | 2 | 100% |
| 6 | Bhattacharjee, S. and Müller, H.-G (2023) Single index fréchet regression | 0.644 | 2 | 2 | 100% |
| 7 | Fan, J. and Yao, Q (1998) Efficient estimation of conditional variance functions in stochastic regression | 0.644 | 2 | 2 | 100% |
| 8 | Hahn, J., Todd, P., and Van der Klaauw, W (2001) Identification and estimation of treatment effects with a regression-discontinuity design | 0.644 | 2 | 2 | 100% |
| 9 | Thistlethwaite, D. L. and Campbell, D. T (1960) Regression-discontinuity analysis: An alternative to the ex post facto experiment | 0.644 | 2 | 2 | 100% |
| 10 | Chow, G. C (1960) Tests of equality between sets of coefficients in two linear regressions | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 60 scored citations.