Checking the adequacy of the multivariate semiparametric location shift model

Let X,X1,...,Xm,..., Y,Y1,...,Yn,... be independent d-dimensional random vectors, where the Xj are i.i.d. copies of X, and the Yk are i.i.d. copies of Y. We study a class of consistent tests for the hypothesis that Y has the same distribution as X+[mu] for some unspecified . The test statistic L is a weighted integral of the squared modulus of the difference of the empirical characteristic functions of and Y1,...,Yn, where is an estimator of [mu]. An alternative representation of L is given in terms of an L2-distance between two nonparametric density estimators. The finite-sample and asymptotic null distribution of L is independent of [mu]. Carried out as a bootstrap or permutation procedure, the test is asymptotically of a given size, irrespective of the unknown underlying distribution. A large-scale simulation study shows that the permutation procedure performs better than the bootstrap.