A High-Dimensional Cramér–von Mises Test

The Cramér–von Mises test provides a useful criterion for assessing goodness of fit in various problems. In this paper, we introduce a novel Cramér–von Mises-type test for testing distributions of high-dimensional continuous data. We establish an asymptotic theory for the proposed test statistics based on quadratic functions in high-dimensional stochastic processes. To estimate the limiting distribution of the test statistic, we propose two practical approaches: a plug-in calibration method and a subsampling method. Theoretical justifications are provided for both techniques. Numerical simulation also confirms the convergence of the proposed methods.