Subsampling inference in cube root asymptotics with an application to Manski’s maximum score estimator

Kim and Pollard (Annals of Statistics, 18 (1990) 191–219) showed that a general class of M-estimators converge at rate n1/3 rather than at the standard rate n1/2. Many times, this situation arises when the objective function is non-smooth. The limiting distribution is the (almost surely unique) random vector that maximizes a certain Gaussian process and is difficult to analyze analytically. In this paper, we propose the use of the subsampling method for inferential purposes. The general method is then applied to Manski’s maximum score estimator and its small sample performance is highlighted via a simulation study.