On asymptotic equivalence of the NPMLE of a monotone density and a Grenander-type estimator in multi-sample biased sampling models

Abstract

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.