Open Access
2021 On asymptotic equivalence of the NPMLE of a monotone density and a Grenander-type estimator in multi-sample biased sampling models
Kwun Chuen Gary Chan, Hok Kan Ling, Tony Sit, Sheung Chi Phillip Yam
Author Affiliations +
Electron. J. Statist. 15(1): 2876-2904 (2021). DOI: 10.1214/21-EJS1812


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.


The first author acknowledges the support by US National Institutes of Health Grant R01HL122212 and US National Science Foundation Grant DMS1711952. The second author thanks for the hospitality at the Chinese University of Hong Kong during the final draft of this work. The third author acknowledges the financial support of HKSAR GRF Grants 14317716 and 14301618. The fourth author acknowledges the financial support from HKGRF- Project Number 14300319 with the project title: Shape-constrained Inference: Testing for Monotonicity, 14300717 with the project title: New Kinds of Forward-Backward Stochastic Systems with Applications. He also thanks Columbia University for the kind invitation to be a visiting faculty member in the Department of Statistics during his sabbatical leave. Phillip Yam also recalls the unforgettable moments and the happiness shared with his beloved father during the drafting of the present article at their home. Although he lost his father with the deepest sadness at the final stage of the review of this work, his father will never leave the heart of Phillip Yam; and he used this work in memory of his father’s brave battle against liver cancer.


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Kwun Chuen Gary Chan. Hok Kan Ling. Tony Sit. Sheung Chi Phillip Yam. "On asymptotic equivalence of the NPMLE of a monotone density and a Grenander-type estimator in multi-sample biased sampling models." Electron. J. Statist. 15 (1) 2876 - 2904, 2021.


Received: 1 June 2020; Published: 2021
First available in Project Euclid: 25 May 2021

Digital Object Identifier: 10.1214/21-EJS1812

Primary: 62G07 , 62G20

Keywords: Argmax continuous mapping theorem , ‎asymptotic ‎equivalence , Density estimation , Karush-Kuhn-Tucker conditions , Shape-constrained estimation , s-sample biased sampling

Vol.15 • No. 1 • 2021
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