## On rates and limit distributions

### J. Pfanzagl

*Mathematical Institute, University of Cologne, Albertus-Magnus-Platz,*

D - 50923 Cologne, Germany
(Received June 16, 1997; revised August 17, 1998)

**Abstract.**
For regular parametric models,
estimators converge uniformly at a rate *n*^{-1/2}, and the
limit distribution is normal with mean 0. The situation is
different if the best possible rate is *n*^{-alpha}, with *alpha* \in (0,1/2), as common for nonparametric models. In
this case, uniformly attainable normal limit distributions
with mean 0 are impossible.

*Key words and phrases*:
Estimation, rates of
convergence, asymptotic optimality.

**Source**
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