Joint modeling of a large number of variables often requires dimension
reduction strategies that lead to structural assumptions of the underlying
correlation matrix, such as equal pair-wise correlations within subsets of
variables. The underlying correlation matrix is thus of interest for both model
specification and model validation. In this paper, we develop tests of the
hypothesis that the entries of the Kendall rank correlation matrix are linear
combinations of a smaller number of parameters. The asymptotic behavior of the
proposed test statistics is investigated both when the dimension is fixed and
when it grows with the sample size. We pay special attention to the restricted
hypothesis of partial exchangeability, which contains full exchangeability as a
special case. We show that under partial exchangeability, the test statistics
and their large-sample distributions simplify, which leads to computational
advantages and better performance of the tests. We propose various scalable
numerical strategies for implementation of the proposed procedures, investigate
their behavior through simulations and power calculations under local
alternatives, and demonstrate their use on a real dataset of mean sea levels at
various geographical locations