Distribution-free testing of linear type
Abstract
We introduce a distribution-free goodness-of-fit test, termed the omega-1 test, which naturally complements the Kolmogorov--Smirnov test and Cramér--von Mises test and can be viewed as their (piecewise) linear analog. Defined as an L1-functional of the empirical process, the test statistic improves on balancing sensitivity to localized and diffuse alternatives and gives a robust and interpretable measure of distributional discrepancy, apart from close connections to the Wasserstein 1-distance. For finite samples, we derive a finite-dimensional computational form for the statistic under general conditions, which leads to various explicit formulas for its null distribution. Under mild continuity assumptions, the limiting statistic is distribution-free, with explicit distribution formulas. In composite settings, the statistic is also compatible with the Khmaladze transformation, enabling asymptotically distribution-free testing. The limiting transformed statistic also has an explicit distribution that escapes reliance on intractable compensator processes or purely numerical evaluation. Simulation results indicate rapid convergence of the finite-sample distributions to their limiting counterparts and support the practical applicability of the test.
Related
- Goodness-of-fit testing for the Pareto type-I distribution based on a mean residual life characterization
- Testing Conditional Distribution Equality Using Generative Models
- Bayes Factor Testing of Linear and Nonlinear Effects in Additive Gaussian Process Regression
- Testing for point-impact effects in a spatial semi-functional linear regression model
- Revisiting dependence in multiple testing: empirical distribution approaches for FDP control
BibTeX
@misc{xia2026distributionfree,
title = {{Distribution-free testing of linear type}},
author = {Weixuan Xia},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30074},
eprint = {2608.30074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30074},
}
RIS
TY - UNPB
TI - Distribution-free testing of linear type
AU - Weixuan Xia
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30074
UR - https://arxiv.org/abs/2608.30074
AB - We introduce a distribution-free goodness-of-fit test, termed the omega-1 test, which naturally complements the Kolmogorov--Smirnov test and Cramér--von Mises test and can be viewed as their (piecewise) linear analog. Defined as an L^1-functional of the empirical process, the test statistic improves on balancing sensitivity to localized and diffuse alternatives and gives a robust and interpretable measure of distributional discrepancy, apart from close connections to the Wasserstein 1-distance. For finite samples, we derive a finite-dimensional computational form for the statistic under general conditions, which leads to various explicit formulas for its null distribution. Under mild continuity assumptions, the limiting statistic is distribution-free, with explicit distribution formulas. In composite settings, the statistic is also compatible with the Khmaladze transformation, enabling asymptotically distribution-free testing. The limiting transformed statistic also has an explicit distribution that escapes reliance on intractable compensator processes or purely numerical evaluation. Simulation results indicate rapid convergence of the finite-sample distributions to their limiting counterparts and support the practical applicability of the test.
ER -