Three monotone density estimators from selection biased samples
Consider the estimation of a nonincreasing probability density function f of a nonnegative random variable based on a sample from a weighted distribution G whose probability density function is given by g(x) = w(x)/w f (x), where w(.) > 0 and w = integral(0)(infinity) w(x)v(x)dx is the normalizing constant. In this paper we give two new estimators one based on the slopes of the least concave majorant of (F) over cap(n), the nonparametric m.l.e of F, and the other based on a penalized likelihood introduced in Woodroofe and Sun (1993). Our estimators are compared using simulations and bootstrap joint confidence bands for f are developed.
density estimation weighted distributions least concave majorant least squares projection maximum likelihood estimation order restricted inference bootstrap confidence bands