A Berry-Esseen Type Bound for a Smoothed Version of Grenander Estimator

Abstract

In various statistical model, such as density estimation and estimation of regression curves or hazard<br /> rates, monotonicity constraints can arise naturally. A frequently encountered problem in nonparametric<br /> statistics is to estimate a monotone density function f on a compact interval. A known estimator for<br /> density function of f under the restriction that f is decreasing, is Grenander estimator, where is the left<br /> derivative of the least concave majorant of the empirical distribution function of the data. Many authors<br /> worked on this estimator and obtained very useful properties from this estimator. Grenander estimator<br /> is a step function and as a consequence it is not smooth. In this paper, we discuss the estimation of a<br /> decreasing density function by the kernel smoothing method. Many works have been done due to the<br /> importance and applicability of Berry-Esseen bounds for the density estimator. In this paper, we study<br /> a Berry- Esseen type bound for a smoothed version of Grenander estimator.