This paper discusses several lower bound results for the asymptotic performance of estimators of smooth functionals in i.i.d. models. The key idea is to look at a set of local limiting distributions of an estimator sequence, rather than to impose regularity conditions, or to consider limits of maximum risk. Special attention is paid to situations where the tangent cone is not a linear space. As an example, the local asymptotic minimax risk in mixture models is computed.
"On the Asymptotic Information Bound." Ann. Statist. 17 (4) 1487 - 1500, December, 1989. https://doi.org/10.1214/aos/1176347377