## Hellinger-consistency of certain nonparametric maximum likelihood estimators.(English)Zbl 0779.62033

Let $$X_ 1,X_ 2,\dots$$ be an i.i.d. sequence from some density $$f_{\theta_ 0}\in\{f_ \theta: \theta\in\Theta\}$$, and denote with $$\theta_ 0$$ the MLE of $$\theta_ 0$$. It is shown that under certain entropy conditions on the class $$\{f_ \theta\}$$, $$f_{\theta_ n}\to f_{\theta_ 0}$$ w.r.t. the Hellinger distance. This is achieved by a bound, which relates the Hellinger distance to the sup of an empirical process indexed by proper transforms of $$f_ \theta$$, $$\theta\in\Theta$$, and then applying a uniform SLLN for empirical d.f.’s.
Reviewer: W.Stute (Gießen)

### MSC:

 62G05 Nonparametric estimation 62G30 Order statistics; empirical distribution functions 60G50 Sums of independent random variables; random walks 62F12 Asymptotic properties of parametric estimators 60F15 Strong limit theorems
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