Classical Laplace estimation for n3-consistent estimators: Improved convergence rates and rate-adaptive inference

We propose a classical Laplace estimator alternative for a large class of n3-consistent estimators, including isotonic regression, monotone hazard, and maximum score estimators. The proposed alternative provides a unified method of smoothing; easier computation is a byproduct in the maximum score case. Depending on input parameter choice and smoothness, the convergence rate of our estimator varies between n3 and (almost) n and its limit distribution varies from Chernoff to normal. We provide a bias reduction method and an inference procedure which automatically adapts to the correct convergence rate and limit distribution.