Statistica Sinica

L. Baringhaus and C. Franz

Abstract:New rigid motion invariant tests for the multivariate two-sample problem are proposed. The test statistic is based on the inter-point distances between the two samples and the inter-point distances within each sample. The asymptotic null distribution of the test statistic is a weighted sum of squares of independent unit normal random variables, the weights being the eigenvalues of a certain Hilbert-Schmidt-operator depending on the unknown underlying distribution. An estimate of the limit distribution is obtained by replacing the unknown weights by the eigenvalues of a bootstrapped version of the operator. Quantiles of the estimate are chosen as critical values. The tests are shown to be consistent. Approximate Bahadur efficiencies computed for normal location alternatives, normal scale alternatives, and Lehmann's contaminated alternative are seen to coincide locally with Pitman efficiencies. The results are supported by a simulation study.

Key words and phrases:Bootstrap in the limit, Cramér test, multivariate two-sample tests, rigid motion invariance.