Document Type : Research Manuscript


Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Thehran, Iran.


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, where is the left
derivative of the least concave majorant of the empirical distribution function of the data. Many authors
worked on this estimator and obtained very useful properties from this estimator. Grenander estimator
is a step function and as a consequence it is not smooth. In this paper, we discuss the estimation of a
decreasing density function by the kernel smoothing method. Many works have been done due to the
importance and applicability of Berry-Esseen bounds for the density estimator. In this paper, we study
a Berry- Esseen type bound for a smoothed version of Grenander estimator.


Main Subjects