  Testing hypotheses of goodness-of-fit about mixture distributions on the
basis of independent but not necessarily identically distributed random vectors
is considered. The hypotheses are given by a specific distribution or by a
family of distributions. Moreover, testing hypotheses formulated by Hadamard
differentiable functionals is discussed in this situation, in particular the
hypothesis of central symmetry, homogeneity and independence.
Kolmogorov-Smirnov or Cram\'er-von-Mises type statistics are suggested as well
as methods to determine critical values. The focus of the investigation is on
asymptotic properties of the test statistics. Further, outcomes of simulations
for finite sample sizes are given. Applications to models with not identically
distributed errors are presented. The results imply that the tests are of
asymptotically exact size and consistent.
