We propose inference tools to analyse the ordering of concordance of random vectors. The analysis in the bivariate case relies on tests for upper and lower quadrant dominance of the true distribution by a parametric or semiparametric model, i.e. for a parametric or semiparametric model to give a probability that two variables are simultaneously small or large at least as great as it would be were they left unspecified. Tests for its generalisation in higher dimensions, namely joint lower and upper orthant dominance, are also analysed. The parametric and semiparametric setting are based on the copula representation for multivariate distribution, which allows for disentangling behaviour of margins and dependence structure. We propose two types of testing procedures for each setting. The first procedure is based on a formuation of the dominance concepts in terms of values taken by random variables, while the second procedure is based on a formulation in terms of probability levels. For each formulation a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US insurance claim data.