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^{−α}, with α ∈ (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.