Abstract
We study the nonparametric estimation of a decreasing density function g (0) in a general s-sample biased sampling model with weight (or bias) functions w(i) for i = 1, …, s. The determination of the monotone maximum likelihood estimator ĝ(n) and its asymptotic distribution, except for the case when s = 1, has been long missing in the literature due to certain non-standard structures of the likelihood function, such as non-separability and a lack of strictly positive second order derivatives of the negative of the log-likelihood function. The existence, uniqueness, self-characterization, consistency of ĝ(n) and its asymptotic distribution at a fixed point are established in this article. To overcome the barriers caused by non-standard likelihood structures, for instance, we show the tightness of ĝ(n) via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the [Formula: see text] -rate of convergence of the linear functional ∫ w(i) ĝ(n).