Robust Estimation of Some Nonregular Parameters

This paper develops optimal estimation of a potentially nondifferentiable functional Г(β) of a regular parameter β, when Г satisfies certain conditions. Primary examples are min or max functionals that frequently appear in the analysis of partially identified models. This paper investigates both the average risk approach and the minimax approach. The average risk approach considers average local asymptotic risk with a weight function Π over β-q(β) for a fixed location-scale equivariant map q, and the minimax approach searches for a robust decision that minimizes the local asymptotic maximal risk. In both approaches, optimal decisions are proposed. Certainly, the average risk approach is preferable to the minimax approach when one has fairly accurate information of β-q(β). When one does not, one may ask whether the average risk decision with a certain weight function Π is as robust as the minimax decision. This paper specifies conditions for Г such that the answer is negative. This paper discusses some results from Monte Carlo simulation studies.