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Donsker-type theorems for nonparametric maximum likelihood estimators. (English) Zbl 1113.60028
Summary: Let $${\mathcal{P}}$$ be a nonparametric probability model consisting of smooth probability densities and let $${\widehat{p}_{n}}$$ be the corresponding maximum likelihood estimator based on $$n$$ independent observations each distributed according to the law $${\mathbb{P}}$$. With $$\widehat{\mathbb{P}}_{n}$$ denoting the measure induced by the density $${\widehat{p}_{n}}$$, define the stochastic process $${\widehat{\nu}}_{n}: f\mapsto \sqrt{n} \int fd({\widehat{\mathbb{P}}}_{n} -\mathbb{P})$$ where $$f$$ ranges over some function class $${\mathcal{F}}$$. We give a general condition for Donsker classes $${\mathcal{F}}$$ implying that the stochastic process $$\widehat{\nu}_{n}$$ is asymptotically equivalent to the empirical process in the space $${\ell^{\infty}(\mathcal{F})}$$ of bounded functions on $${\mathcal{F}}$$. This implies in particular that $$\widehat{\nu}_{n}$$ converges in law in $${\ell^{\infty}(\mathcal{F})}$$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes $${\mathcal{ F}}$$. We give a number of applications: convergence of the probability measure $${\widehat{\mathbb{P}}_{n}}$$ to $${\mathbb{P}}$$ at rate $${\sqrt{n}}$$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; $${\sqrt{n}}$$ -efficient estimation of nonlinear functionals defined on $${\mathcal{P}}$$; limit theorems at rate $${\sqrt{n}}$$ for the maximum likelihood estimator of the convolution product $${\mathbb P\ast \mathbb P}$$.

##### MSC:
 60F05 Central limit and other weak theorems 62G07 Density estimation 62F12 Asymptotic properties of parametric estimators 46F05 Topological linear spaces of test functions, distributions and ultradistributions
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