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