A new class of goodness-of-fit tests using weighted degenerate U-statistics

Abstract

A novel flexible class of goodness-of-fit tests is introduced for univariate and multivariate data based on weighted degenerate U-statistics. The test statistics are constructed as linear combinations of U-statistics, with weights chosen to standardize the contribution of each component. We derive their asymptotic distributions under the null hypothesis and evaluate their general performance through simulations. As concrete examples, we demonstrate the method’s application to testing multivariate normality and von Mises distribution on the circle, showing competitive power against existing alternatives while maintaining proper control of empirical size.