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 wi for i = 1, …, s. The determination of the monotone maximum likelihood estimator ĝn 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 ĝn 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 ĝn via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the -rate of convergence of the linear functional ∫ wi ĝn.
Keywords: Karush-Kuhn-Tucker conditions; density estimation; empirical process theory; nonparametric estimation; order statistics from multiple samples; s-sample biased sampling; self-induced characterization; shape-constrained problem.