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We consider in this paper two widely studied examples of nonparametric and semiparametric models in which the standard information bounds are totally misleading. In fact, no estimators converge at the n^- rate for any > 0, although the information is strictly positive "promising" that n^-1/2 is achievable. The examples are the estimation of p² and the slope in the model of Engle et al. A class of models in which the parameter of interest can be estimated efficiently is discussed.
Ritov et al. (Fri,) studied this question.