**Bernard Omolo**
**Texas Tech
University**
**Abstract**

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 and the alternatives are local.

We apply the
Chernoff - Savage approach to derive the asymptotic distribution of a test
statistic for a

repeated
observations
model with an orthonormal design, given that the centered design matrices
are not

of full rank.
The distribution turns out to be chi-square under the null hypothesis,
regardless of the

choice of the
aligner. Simulation studies regarding the Type I error rate and power in
testing for

linearity of
a nonparametric regression model with standard Cauchy random errors corroborate
this

theoretical
result. Additional insight into the independence on the aligner is gained
in scale models.

This approach
can be extended to the multivariate case using a projection
method.