A K-sample testing problem is studied for multivariate semi-Markov counting
processes. Asymptotic distributions and efficiency of a class of nonparame
tric test statistics are established for certain local alternatives. The co
ncept of the asymptotic efficiency stales that for every nonparametric test
in this class, there is a parametric submodel for which the optimal test h
as the same asymptotic power as the nonparametric test. The theory is illus
trated by a simulation study and by analyzing the multivariate failure time
data of Thompson et al. (1978).