We show that the Watson and Cramér-von Mises statistics are related to Green's function of the Laplacian on a circle. A generalization leads to a new U-statistic whose kernel is the Green function of the Laplacian on the sphere. © 2007 Elsevier B.V. All rights reserved.
Pycke, J. R. (2007). U-statistics based on the Green’s function of the Laplacian on the circle and the sphere. Statistics and Probability Letters, 77(9), 863–872. https://doi.org/10.1016/j.spl.2006.11.009