We propose a way to construct consistent generalizations of the Cram\'er-von Mises test on the line, and the Watson $U_n^{\,2}$ test on the circle, based on classes of partitions invariant under various group actions. The framework developed extends our earlier work generalizing the Kolmogorov-Smirnov and Kuiper goodness-of-fit tests, and provides a conceptually unifying description. Many goodness-of-fit tests for uniformity on the circle incorporate rotation invariance. Our construction does not require that the distribution for the null hypothesis be uniform, and permits tests invariant under general coordinate transformations. Some distribution-free tests are presented as examples, and their properties explored through numerical simulations.
Key words and phrases:
Distribution-free tests; Kolmogorov-Smirnov test;
Kuiper's test; Cram\'er-von Mises test;
Watson $U_n^{\,2}$ test.
Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1998/98-45.ps.gz
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