Classical Laplace estimation for cube-root-n-consistent estimators: Improved convergence rates and rate-adaptive inference

Abstract

We propose a classical Laplace estimator alternative for a large class of $\sqrt[3]{n}$-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 $\sqrt[3]{n}$ and (almost) $\sqrt{n}$ 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.