  A two-class mixture model, where the density of one of the components is
known, is considered. We address the issue of the nonparametric adaptive
estimation of the unknown probability density of the second component. We
propose a randomly weighted kernel estimator with a fully data-driven bandwidth
selection method, in the spirit of the Goldenshluger and Lepski method. An
oracle-type inequality for the pointwise quadratic risk is derived as well as
convergence rates over Holder smoothness classes. The theoretical results are
illustrated by numerical simulations.
