We study the nonparametric estimation of a decreasing density function g0 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 n-rate of convergence of the linear functional ∫ wi ĝn,

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In this article, we show that the nonparametric maximum likelihood estimator (NPMLE) of the decreasing density function in the s-sample biased sampling models, is asymptotically equivalent to a Grenander-type estimator, namely the left-continuous slope of the least concave majorant of the NPMLE of the distribution function in the larger model without imposing the monotonicity assumption. Since the two estimators favor different proof directions in establishing weak convergence, we require additional results for both estimators so that the two estimators can be considered jointly in a unified approach. For instance, we employ an analytic argument for showing the tightness of an inverse processes associated with the NPMLE, since a conventional geometric approach used in the literature cannot be employed due to multiple biased samples. We demonstrate other results using numerical simulation and a real data illustration.