Bivariate and multivariate sign depth and related distribution-free tests for model fit

For testing a model fit for univariate data, tests based on the K -sign depth introduced by Malcherczyk et al. (2021) are distribution-free, outlier robust and very powerful for $$K > 2$$ K > 2 . Here, we propose two extensions of the K -sign depth to p -dimensional data, namely the L -simplex depth based on L simplices spanned by $$p + 1$$ p + 1 residuals and component L -depth, which is the componentwise application of the K -sign depth with $$K=L+1$$ K = L + 1 . The simplex depth as well as the component depth can be used in a full version and in a simplified version. We derive some properties of the two simplex depth versions for bivariate data which show in particular that tests based on these depth versions are distribution-free. Then, we compare the four depth versions with $$L = 1$$ L = 1 and $$L = 2$$ L = 2 by simulations. We also provide a simple algorithm to calculate all component depths in linear time and efficient algorithms for the simplex depths.