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Aligned rank statistics for repeated measurement models with orthonormal design, employing a Chernoff–Savage approach
Authors:JHJ Einmahl  BO Omolo  ML Puri  FH Ruymgaart
Institution:

aDepartment of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

bDepartment of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA

cDepartment of Mathematics, Indiana University, Rawles Hall 232, Bloomington, IN 47405, USA

Abstract:In this paper, aligned rank statistics are considered for testing hypotheses regarding the location in repeated measurement designs, where the design matrix for each set of measurements is orthonormal. Such a design may, for instance, be used when testing for linearity. It turns out that the centered design matrix is not of full rank, and therefore it does not quite satisfy the usual conditions in the literature. The number of degrees of freedom of the limiting chi-square distribution of the test statistic under the null hypothesis, however, is not affected, unless rather special hypotheses are tested. An independent derivation of this limiting distribution is given, using the Chernoff–Savage approach. In passing, it is observed that independence of the choice of aligner, which in the location problem is well-known to be due to cancellation, may in scale problems occur as a result of the type of score function suitable for scale tests. A possible extension to multivariate data is briefly indicated.
Keywords:Aligned rank statistics  Orthonormal design matrix  Repeated measurements  Chernoff–Savage approach
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