Abstract:
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.
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