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.

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