Abstract
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.
Highlights
We consider in this paper multidimensional mixture models that describe the probability distribution of a random vector X with at least three coordinates
We prove in Proposition 2 that when the partition is refined, the Fisher information associated to this partition increases. This leads to our main general result presented in Theorem 1, Section 2.3: it is possible to let the approximation parametric models grow with the sample size so that the sequence of maximum likelihood estimators are asymptotically efficient in the semiparametric model and so that a semiparametric Bernstein-von Mises Theorem holds
3 Model selection In Theorem 1, we prove the existence of some increasing partition leading to efficiency
Summary
We consider in this paper multidimensional mixture models that describe the probability distribution of a random vector X with at least three coordinates. In the particular case of the semiparametric mixtures, this is of great interest, since the construction of a confidence region is not necessarily trivial This is our first main result which is stated in Theorem 1: by considering partitions refined slowly enough when the number of observations increases, we can derive efficient estimation procedures for the parameter of interest θ and in the Bayesian approach for a marginal posterior distribution on θ which satisfies the renowned Bernstein-von Mises property. We prove in Proposition 2 that when the partition is refined, the Fisher information associated to this partition increases This leads to our main general result presented in Theorem 1, Section 2.3: it is possible to let the approximation parametric models grow with the sample size so that the sequence of maximum likelihood estimators are asymptotically efficient in the semiparametric model and so that a semiparametric Bernstein-von Mises Theorem holds.
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