
Efficient semiparametric estimation and model selection for multidimensional mixtures
Elodie Vernet, Cambridge University
Obtaining theoretical guarantees (such as uncertainty quantification) in the context of parameter estimation may be challenging in mixture models. Note that identifiability is already not trivial in these models. In this presentation, I will discuss efficiency in the context of nonparametric mixture models. More precisely, we consider mixture models where the i.i.d. observations have at least three components which are independent given the population of the observation. We don't assume a parametric modelling of the emission distributions that is the distribution of the observation given its population. And we are interested in the semiparametric estimation of the proportion of each population. Using a discretisation of the problem via projection of the densities in histograms, we obtain an asymptotically efficient estimator. In the Bayesian setting, using a sequence of prior distributions defined on more and more complex sets when the number of observations increases, we show that the associated sequence of posterior distribution verifies a Bernstein von Mises type theorem with efficient Fisher information for the semiparametric problem as variance. These two asymptotic results are true given the complexity of the approximation models don't increase too fast compared to the number of observations. We then propose a crossvalidation like procedure to select the complexity of the model in a finite horizon. This proposed procedure satisfies an oracle inequality.
These results are part of a joint work with Elisabeth Gassiat and Judith Rousseau. Reference: https://arxiv.org/abs/1607.05430.

