Abstract: The goal of this paper is to study the bootstrap for the Grenander estimator. The first result is a proof of the inconsistency of the nonparametric bootstrap for the Grenander estimator at a given point. The second result is the development and verification of a bootstrap for the L1 confidence band for the Grenander estimator. As part of this work, kernel estimators are studied as alt...

In various statistical model, such as density estimation and estimation of regression curves or hazard rates, monotonicity constraints can arise naturally. A frequently encountered problem in nonparametric statistics is to estimate a monotone density function f on a compact interval. A known estimator for density function of f under the restriction that f is decreasing, is Grenander estimator, ...

In this paper we consider the stacking of isotonic regression and method rearrangement with empirical estimator to estimate a discrete distribution an infinite support. The estimators are proved be strongly consistent n-rate convergence. We obtain asymptotic distributions construct asymptotically correct conservative global confidence bands. show that stacked Grenander outperforms estimator. ne...

In this paper we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate n 1 3 . The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function f on [0,∞), is a prototypical example. We focus on this example and explore different approaches to cons...

We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density f(0) is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations using tools from convex analysis. The theory is applied to a problem involving mixtures.

Based on empirical Levy-type concentration functions a new graphical representation of the ML-density estimator under order restrictions is given. This representation generalizes the well-known representation of the Grenander estimator of a monotone density as the slope of the least concave majorant of the empirical distribution function. From the given representation it follows that a density ...

The first part of this paper gives some general consistency theorems for the maximum product of spacings (MPS) method, an estimation method related to maximum likelihood. The second part deals with nonparametric estimation of a concave (convex) distribution and more generally a unimodal distribution, without smoothness assumptions on the densities. The MPS estimator for a distribution function ...