ESTIMATION OF A MONOTONE DENSITY IN s-SAMPLE BIASED SAMPLING MODELS

• 成果类型: Article
• 署名作者: Chan, Kwun Chuen Gary; Ling, Hok Kan; Sit, Tony; Yam, Sheung Chi Phillip
• 署名单位: University of Washington; University of Washington Seattle; Columbia University; Chinese University of Hong Kong
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1614
• 发表日期: 2018
• 页码: 2125-2152
• 关键词: empirical distributions; likelihood

摘要

We study the nonparametric estimation of a decreasing density function go in a general s-sample biased sampling model with weight (or bias) functions w(i )for i = 1, ...,s. The determination of the monotone maximum likelihood estimator (g) over cap (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 (g) over cap (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 (g) over cap (n) via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the root n-rate of convergence of the linear functional integral w(i )(g) over cap (n).