  The problem of testing the equality of the generating processes of two
categorical time series is addressed in this work. To this aim, we propose
three tests relying on a dissimilarity measure between categorical processes.
Particular versions of these tests are constructed by considering three
specific distances evaluating discrepancy between the marginal distributions
and the serial dependence patterns of both processes. Proper estimates of these
dissimilarities are an essential element of the constructed tests, which are
based on the bootstrap. Specifically, a parametric bootstrap method assuming
the true generating models and extensions of the moving blocks bootstrap and
the stationary bootstrap are considered. The approaches are assessed in a broad
simulation study including several types of categorical models with different
degrees of complexity. Advantages and disadvantages of each one of the methods
are properly discussed according to their behavior under the null and the
alternative hypothesis. The impact that some important input parameters have on
the results of the tests is also analyzed. An application involving biological
sequences highlights the usefulness of the proposed techniques.
