Generalized multivariate rank type test statistics via spatial U-quantiles

The classical univariate sign and signed rank tests for location have been extended over the years to the multivariate setting, including recent robust rotation invariant "spatial" versions. Here we introduce a broad class of rotation invariant multivariate spatial generalized rank type test statistics. For a given inference problem not restricted to location, the test statistics are linked through Bahadur-Kiefer representations with spatial median estimators in appropriately matched U-quantile location models. Under null and contiguous alternative hypotheses, related quadratic form statistics have central and noncentral chi-square limit distributions. Robustness properties in terms of breakdown points and influence functions of the associated estimators are quite favorable. Illustrative applications cover location, multivariate dispersion, and multiple linear regression.