The montone density estimators from selection biased samples

Abstract

Consider the estimation of a nonincreasing probability density function f of a nonnegativerandom variable based on a sample from a weighted distribution G whose probability density function is given by

where w (·) > 0 and is the normalizing constant. In this paper y egive two new estimators off, one based on the slopes of the least concave majorant of [Fcirc]_n the nonparametric m.1.e of F, and the other based on a penalized likelihood introducedin Woodroofe and Sun (1993). Our estimators are compared using simulations andbootstrap joint confidence bands for f are developed.

Keywords:

• Density estimation
• Least concave majorant
• Least squares projection
• Maximum likelihood estimation
• Order restricted inferenc
• Weighted distributions
• Bootstrap coddence bands