Local asymptotic minimax estimation of nonregular parameters with translation-scale equivariant maps

When a parameter of interest is de…ned to be a nondi¤erentiable transform of a regular parameter, the parameter does not have an in‡uence function, rendering the existing theory of semiparametric e¢ cient estimation inapplicable. However, when the nondi¤erentiable transform is a known composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map, this paper demonstrates that it is possible to de…ne a notion of asymptotic optimality of an estimator as an extension of the classical local asymptotic minimax estimation. This paper establishes a local asymptotic risk bound and proposes a general method to construct a local asymptotic minimax decision.