On Estimated Projection Pursuit-Type Crámer–von Mises Statistics

This paper addresses the problem of testing for a multivariate distribution, which belongs to a known parametric distribution family. The estimated Crámer–Von Mises-type test statistics are constructed using projection pursuit technique. Some interested properties of the test statistics, like asymptotics, bootstrap approximations, and the tail behavior of the limits of test statistics are investigated. For computational reasons, an approximation via the number theoretic method to the extreme value and the integral on a super sphere surface is considered.