In this paper, we consider nonparametric multidimensional finite mixture
models and we are interested in the semiparametric estimation of the population
weights. Here, the i.i.d. observations are assumed to have at least three
components which are independent given the population. We approximate the
semiparametric model by projecting the conditional distributions on step
functions associated to some partition. Our first main result is that if we
refine the partition slowly enough, the associated sequence of maximum
likelihood estimators of the weights is asymptotically efficient, and the
posterior distribution of the weights, when using a Bayesian procedure,
satisfies a semiparametric Bernstein von Mises theorem. We then propose a
cross-validation like procedure to select the partition in a finite horizon.
Our second main result is that the proposed procedure satisfies an oracle
inequality. Numerical experiments on simulated data illustrate our theoretical
results