ELSEVIER Statistics & Probability Letters 36 (1997) 9-21 STATISTII~ & Studentized permutation tests for non-i.i.d, hypotheses and the generalized Behrens-Fisher problem Arnold Janssen* Mathematical lnstitut, University of Dfisseldorf Universitiitsstrasse1, D-40225 Diisseldorf Germany Received 1 December 1995; receivedin revised form 1 December 1996 Abstract It is shown that permutation tests based on studentized statistics are asymptotically exact of size ~ also under certain extended non-i.i.d, null hypotheses. To demonstrate the principle the results are applied to the generalized two-sample Behrens-Fisher problem for testing equality of the means under general non-parametric heterogeneous error distribu- tions. Within this setting we propose a permutation version of the Welch test which is an extension of Pitman's two-sample permutation test. These results are special cases of a conditional central limit theorem for studentized permutation statistics which also applies to asymptotic power functions. © 1997 Elsevier Science B.V. AMS classification: 62G 10; 62G09 Keywords: Behrens-Fisher problem; Permutation test; Permutation statistic; Conditional central limit theorem; Survival test 1. Introduction and examples Throughout, we are concerned with a two-sample non-parametric testing problem given by an extended null hypothesis Ho which is strictly larger than the restricted null hypothesis of i.i.d, random variables. The restriction of Ho to the i.i.d, case is denoted by iZlo, i.e. 1-7Io ~ Ho. A famous example is the extended Behrens-Fisher problem where equality of two means is tested for a model with different unknown variances of the error variables. For this type of examples the choice of critical values (under Ho) is a serious problem. As message of this paper we suggest to carry out the underlying tests as permutation tests for studentized statistics which includes a variance correction of the permutation distribution. It is shown that in this case the conditional critical values of the permutation distribution work well at least in the asymptotic setting. We follow the approach of Neuhaus (1993), who successfully applied studentized permutation tests for survival problems, see Example 1.2 below. * Tel.: + 49 21181 12165; fax: + 49 211 81 13117; e-mail:janssena@clio.rz.uni-duesserdorf.de. 0167-7152/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S0 1 67-7 1 52(97)00043-6