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+μ for some unspecified μ∈Rd. The test statistic L is a weighted integral of the squared modulus of the difference of the empirical characteristic functions of X1+μ̂,…,Xm+μ̂ and Y1,…,Yn, where μ̂ is an estimator of μ. 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 μ. 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.