Testing hypotheses about mixture distributions using not identically distributed data

Testing goodness-of-fit with simple hypotheses and hypotheses of families of distributions about mixture distributions on the basis of observations of independent but not necessarily identically distributed random vectors is discussed. Moreover, testing hypotheses formulated by Hadamard differentiable functionals, e.g., hypothesis of central symmetry, homogeneity and independence, is considered in this model. Kolmogorov-Smirnov or Cram\'er-von-Mises type statistics and bootstrap Monte-Carlo procedures are suggested. In addition, asymptotic properties of the test statistics are investigated. The results imply the asymptotic exactness and the consistency of the tests. Further, outcomes of simulations are given.