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.
Highlights
Biased sampling problems have long been an important issue in a wide array of scientific studies
The main goal in the present article is to show that gn and gn in s-sample biased sampling models are asymptotically equivalent in the sense that n1/3[gn(t0)−gn(t0)] converges to 0 in probability as n → ∞
This paper establishes the asymptotic equivalence between the nonparametric maximum likelihood estimate (NPMLE) of the decreasing density function and the Grenander-type estimator without any shape constraint under the s-sample biased sampling models considered in [3]
Summary
Biased sampling problems have long been an important issue in a wide array of scientific studies. One may not expect this kind of correspondence to hold exactly, but perhaps just asymptotically; for instance, in the random right censorship model, [14] showed that the NPMLE of a decreasing density function is asymptotically equivalent to the left-continuous slope of the Kaplan-Meier estimator, the NPMLE of the distribution function in the absence of the monotonicity assumption. A similar result was shown in [14] to hold for the NPMLE of a decreasing hazard rate, where the NPMLE is asymptotically equivalent to the estimator obtained as the left-continuous slope of the least concave majorant of the Nelson-Aalen estimator, i.e., the NPMLE of the cumulative hazard function without the monotonicity assumption. [18] showed that similar results are true in the Cox model; the NPMLE of an increasing baseline hazard and the left-hand slope of the greatest convex minorant of the Breslow estimator are asymptotically equivalent.
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