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)


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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