  This paper deals with statistical tests on the components of mixture
densities. We propose to test whether the densities of two independent samples
of independent random variables $Y_1, ..., Y_n$ and $Z_1, ..., Z_n$ result from
the same mixture of $M$ components or not. We provide a test procedure which is
proved to be asymptotically optimal according to the minimax setting. We
extensively discuss the connection between the mixing weights and the
performance of the testing procedure and illustrate it with numerical examples.
