Nonparametric symmetry tests for statistical functionals

Dedicated to the 70th birthday of Johann Pfanzagl Summary. Along the lines of Pfanzagl's work the testing theory for the non-parametric null hypothesis of symmetry (including matched pairs) is developed. The testing problem is typically given by a skew symmetric statistical functional which seems to be adequate for the nonparametric world. Under mild regularity assumptions asymptotically maximin tests are obtained which turn out to be eecient for special submodels. The tests can be carried out as nite sample distribution free randomization tests conditioning under the absolute values. It is shown that these tests are asymptotically valid level tests also under some extended null hypotheses when the random variables have heterogeneous distributions under the null hyptohesis. In the case of invariant statistical functionals we end up with rank tests for matched pairs. These rank tests can now be derived by testing problems for functionals.