Open Access
October 2018 Estimation of a monotone density in $s$-sample biased sampling models
Kwun Chuen Gary Chan, Hok Kan Ling, Tony Sit, Sheung Chi Phillip Yam
Ann. Statist. 46(5): 2125-2152 (October 2018). DOI: 10.1214/17-AOS1614


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 $w_{i}$ for $i=1,\ldots,s$. The determination of the monotone maximum likelihood estimator $\hat{g}_{n}$ and its asymptotic distribution, except for the case when $s=1$, has been long missing in the literature due to certain nonstandard structures of the likelihood function, such as nonseparability and a lack of strictly positive second order derivatives of the negative of the log-likelihood function. The existence, uniqueness, self-characterization, consistency of $\hat{g}_{n}$ and its asymptotic distribution at a fixed point are established in this article. To overcome the barriers caused by nonstandard likelihood structures, for instance, we show the tightness of $\hat{g}_{n}$ via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the $\sqrt{n}$-rate of convergence of the linear functional $\int w_{i}\hat{g}_{n}$.


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Kwun Chuen Gary Chan. Hok Kan Ling. Tony Sit. Sheung Chi Phillip Yam. "Estimation of a monotone density in $s$-sample biased sampling models." Ann. Statist. 46 (5) 2125 - 2152, October 2018.


Received: 1 February 2016; Revised: 1 May 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964328
MathSciNet: MR3845013
Digital Object Identifier: 10.1214/17-AOS1614

Primary: 62E20 , 62G20
Secondary: 62G08

Keywords: $s$-sample biased sampling , Density estimation , Empirical process theory , Karush–Kuhn–Tucker conditions , nonparametric estimation , order statistics from multiple samples , self-induced characterization , shape-constrained problem

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
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