Local Asymptotic Minimax Estimation of Nonregular Parameters with Translation-Scale Equivariant Maps

When a parameter of interest is defined to be a nondifferentiable transform of a regular parameter, the parameter does not have an influence function, rendering the existing theory of semiparametric efficient estimation inapplicable. However, when the nondifferentiable 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 define 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.