32 research outputs found
Karhunen-Lo\`eve expansion for a generalization of Wiener bridge
We derive a Karhunen-Lo\`eve expansion of the Gauss process , , where is a
standard Wiener process and is a twice continuously
differentiable function with and . This
process is an important limit process in the theory of goodness-of-fit tests.
We formulate two special cases with the function
, , and , ,
respectively. The latter one corresponds to the Wiener bridge over from
to .Comment: 25 pages, 1 figure. The appendix is extende
U-statistics based on the Green's function of the Laplacian on the circle and the sphere
We show that the Watson and Cramer-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.Directional statistics Tests of uniformity U- and V-statistics Cramer-von Mises statistic Anderson-Darling statistic Watson statistic