Testing a multivariate distribution for generalized skew ellipticity

We consider the problem of testing whether a sample comes from a family of the multivariate generalized skew-elliptical distributions with an unknown location parameter, an unknown scaling matrix, and an unknown distribution of the symmetric component, specified up to a parameter skewing function with an unknown parameter value. We propose test statistics that are functionals of empirical processes indexed by classes of functions. Under mild smoothness conditions on the skewing function and the functional class, we obtain the asymptotic theory for these tests. They are consistent against any fixed alternative, invariant under a group of affine transformations, and flexible to implement. However, the limiting process depends on the unknown parameters in a complicated way. To overcome this obstacle, we propose a bootstrapped modification of the testing procedure, prove that it works theoretically, and illustrate its practical performance on a simulation study.