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 Cram\'{e}r--von Mises test and can be viewed as their (piecewise) linear analog. Defined as an $\mathrm{L}^{1}$-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.