Aligned rank tests for repeated observation models with orthonormal design

Rank tests are known to be distribution-free for simple linear models, where the observations are i.i.d. For general linear models with nuisance parameters, however, the alignment principle can be applied to obtain asymptotically distribution-free rank tests. This is especially so when the centered design matrices have full rank, as is the case in Kraft and van Eeden [19], Adichie [1] and Chiang and Puri [4], among others. Motivated by the example of testing linearity in a nonparametric regression model, however, we will be dealing with models whose centered design matrices are not of full rank.

More specifically, the asymptotic distribution of the aligned rank statistics will be obtained under the null hypothesis and local alternatives for testing a linear hypothesis in a repeated observation model with orthonormal design matrix. These asymptotic distributions are of the chi-square type and independent of the choice of aligner, as in the full rank case. Some simulations of the power function when the errors have a Cauchy distribution are included.

The theory is presented in a self-contained manner, and based on the Chernoff- Savage rather than the Hajek approach. In principle this would allow us 1o also deal with the asymptotics under fixed alternatives, although this option is not presented to completion.

This approach can be extended to scale models as well as multivariatc models. Although these topics are only briefly considered, interesting additional insight, is gained in the independence of the aligner. In the location model, this independence is obtained due to a suitable choice of the test statistic so that cancellation takes place. In the scale model, on the other hand, the aligner docs not e\en appear in the expansion of the basic components of the test statistic due to the particular form of scene functions employed for scale problems.