  In this work, nonparametric log-rank-type statistical tests are introduced in
order to verify homogeneity of purely discrete variables subject to arbitrary
right-censoring for infinitely many categories. In particular, the Cram\'er-von
Mises test statistics for discrete models under censoring is established. In
order to introduce the test, we develop the weighted log-rank statistics in a
general multivariate discrete setup which complements previous fundamental
results of Gill (1980) and Andersen et al. (1982). Due to the presence of
persistent jumps over the unbounded set of categories, the asymptotic
distribution of the test is not distribution-free. The statistical test for a
large class of weighted processes is described as a weighted series of
independent chi-squared variables whose weights can be consistently estimated
and the associated limiting covariance operator can be infinite-dimensional.
The test is consistent to any alternative hypothesis and, in particular, it
allows us to deal with crossing hazard functions. We also provide a simulation
study in order to illustrate the theoretical results.
