(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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