A note on estimating a non-increasing density in the presence of selection bias

In this paper we construct the non-parametric maximum likelihood estimator (NPMLE) (f) over cap (n) of a non-increasing probability density function f with distribution function F on the basis of a sample from a weighted distribution G with density given by g(x) = w(x)f(x)/mu(f,w), where w(u) > 0 for all u and mu(f, w) = integral w(u)f(u)du < infinity is the normalizing constant. We show that the NPMLE of f is proportional to the Grenander (Skand. Akt. 39 (1956) 125) estimator of the density of transformed data using a simple transformation based on w. We explore some of the properties of f. and show that the Prakasa Rao Theorem (Sankhya A 31 (1969) 23) extends to the weighted case. We also give conditions under which the resulting distribution function (F) over cap (n) is strongly uniformly consistent and show that a rate of convergence of order n(-1/2) can be achieved under conditions on w. We also investigate estimation of f when a second sample directly from f is available and carry out a small-scale simulation study of the performance of two estimators in this case. (C) 2002 Elsevier Science B.V. All rights reserved.