We propose new data driven score rank tests for univariate symmetry around a known center.We apply both Schwarz-type and recently introduced data driven penalty selection rules. Some key asymptotic results regarding the test statistics are given and some asymptotic optimality properties proved. In an extensive simulation study, we compare the empirical behaviour of these tests to tests found in the recent literature to be powerful. We show that, for a broad range of asymmetric distributions, data driven tests have stable power, which is comparable to their competitors for typical alternatives and much greater for some atypical alternatives.
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Applying the strong approximation technique we present a unified approach to asymptotic results for multivariate linear rank statistics for the two-sample problem. We reprove asymptotic normality of these statistics under the null hypothesis and under local alternatives convergent at a moderate rate to the null hypothesis. We also provide a moderate deviation theorem for these statistics under the null hypothesis. Proofs are short and use natural argumentation.
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