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The asymptotic risk of efficient estimators with Kullback-Leibler loss in smoothly parametrized statistical models is k/2n, where k is the parameter dimension and n is the sample size. Under fairly general conditions, we given a simple information-theoretic proof that the set of parameter values where any arbitrary estimator is superefficient is negligible. The proof is based on a result of Rissanen that codes have asymptotic redundancy not smaller than (k/2)log n, except in a set of measure 0.

Information theory and superefficiency