n–consistent estimators: improved convergence rates and rate–adaptive inference

We propose a classical (nonBayesian) Laplace estimator alternative for a large class of 3 p n–consistent estimators, including isotonic density and regression estimators, inverse density and regression estimators, the maximum score and mode regression estimators, and interval censoring and monotone hazard rate estimators. The proposed alternative provides a unified method of smoothing that applies to all examples mentioned above; easier computation is a byproduct in the maximum score case. Depending on the choice of input parameter and the degree of smoothness of a population function, the convergence rate of our estimator can be faster than 3 p n and its limit distribution can be normal. With extreme smoothness, a rate close to p n is achievable. We provide a bias reduction method and an inference procedure which automatically adapts to the correct convergence rate and limit distribution.