Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem

Abstract It is shown that permutation tests based on studentized statistics are asymptotically exact of size α 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 distributions. 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.

[1]  A monte carlo study of the studentized wilcoxon statistic for the behrens- fisher problem , 1979 .

[2]  Peter Hall,et al.  ON THE BOOTSTRAP AND TWO-SAMPLE PROBLEMS , 1988 .

[3]  P. Good,et al.  Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses , 1995 .

[4]  On the Behrens-Fisher problem , 1970 .

[5]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[6]  R. Beran.  Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements , 1988 .

[7]  J. Praestgaard.  Permutation and bootstrap Kolmogorov-Smirnov tests for the equality of two distributions , 1995 .

[8]  Hermann Witting,et al.  Mathematische Statistik II , 1985 .

[9]  George E. Policello,et al.  Robust Rank Procedures for the Behrens-Fisher Problem , 1981 .

[10]  G. Neuhaus,et al.  Conditional Rank Tests for the Two-Sample Problem Under Random Censorship , 1993 .

[11]  Welch Bl.  THE GENERALIZATION OF ‘STUDENT'S’ PROBLEM WHEN SEVERAL DIFFERENT POPULATION VARLANCES ARE INVOLVED , 1947 .

[12]  Zhuzhoma Evgeny Victorovich,et al.  Translation of Mathematical Monographs , 1996 .

[13]  J. Pfanzagl.  On the Behrens-Fisher problem , 1974 .

[14]  Joseph P. Romano.  On the behaviour of randomization tests without the group invariance assumption , 1990 .

[15]  Pranab Kumar Sen,et al.  On studentized non-parametric multi-sample location tests , 1962 .

[16]  A. Janssen,et al.  Conditional Rank Tests for Randomly Censored Data , 1991 .

[17]  M. Newton,et al.  A Rank Statistics Approach to the Consistency of a General Bootstrap , 1992 .

[18]  P. Sen,et al.  Theory of rank tests , 1969 .

[19]  B. L. Welch.  THE SIGNIFICANCE OF THE DIFFERENCE BETWEEN TWO MEANS WHEN THE POPULATION VARIANCES ARE UNEQUAL , 1938 .

[20]  Practical recipes to solve the behrens-fisher problem : (prepublication) , 1974 .

[21]  G. Neuhaus.  Asymptotically optimal rank tests for the two-sample problem with randomly censored data , 1988 .

[22]  John C. W. Rayner,et al.  Welch's approximate solution for the Behrens-Fisher problem , 1987 .

[23]  Joseph P. Romano.  Bootstrap and randomization tests of some nonparametric hypotheses , 1989 .

[24]  Local asymptotic normality for randomly censored models with applications to rank tests , 1989 .

[25]  R. Fisher.  THE FIDUCIAL ARGUMENT IN STATISTICAL INFERENCE , 1935 .

[26]  Ying Y. Wang,et al.  Probabilities of the Type I Errors of the Welch Tests for the Behrens-Fisher Problem , 1971 .

[27]  M. Springer,et al.  The Distribution of Products of Independent Random Variables , 1966 .