Bridging Root-$n$ and Non-standard Asymptotics: Adaptive Inference in M-Estimation

Abstract: This manuscript studies a general approach to construct confidence sets for the solution of population-level optimization, commonly referred to as M-estimation. Statistical inference for M-estimation poses significant challenges due to the non-standard limiting behaviors of the corresponding estimator, which arise in settings with increasing dimension of parameters, non-smooth objectives, or constraints. We propose a simple and unified method that guarantees validity in both regular and irregular cases. Moreover, we provide a comprehensive width analysis of the proposed confidence set, showing that the convergence rate of the diameter is adaptive to the unknown degree of instance-specific regularity. We apply the proposed method to several high-dimensional and irregular statistical problems.