LARGE-SAMPLE STUDY OF THE KERNEL DENSITY ESTIMATORS UNDER MULTIPLICATIVE CENSORING

• 成果类型: Article
• 署名作者: Asgharian, Masoud; Carone, Marco; Fakoor, Vahid
• 署名单位: McGill University; University of California System; University of California Berkeley; Ferdowsi University Mashhad
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS954
• 发表日期: 2012
• 页码: 159-187
• 关键词: empirical distributions; strong approximations; BIAS; Consistency; selection; survival

摘要

The multiplicative censoring model introduced in Vardi [Biometrika 76 (1989) 751-761] is an incomplete data problem whereby two independent samples from the lifetime distribution G, X-m = (X-1, ... , X-m) and Z(n) = (Z(1), ... , Z(n)), are observed subject to a form of coarsening. Specifically, sample X-m is fully observed while Y-n = (Y-1, ... , Y-n) is observed instead of Z(n), where Y-i = U(i)Z(i) and (U-1, ... , U-n) is an independent sample from the standard uniform distribution. Vardi [Biometrika 76 (1989) 751-761] showed that this model unifies several important statistical problems, such as the deconvolution of an exponential random variable, estimation under a decreasing density constraint and an estimation problem in renewal processes. In this paper, we establish the large-sample properties of kernel density estimators under the multiplicative censoring model. We first construct a strong approximation for the process root k((G) over cap - G), where (G) over cap is a solution of the nonparametric score equation based on (X-n, Y-n), and k = m + n is the total sample size. Using this strong approximation and a result on the global modulus of continuity, we establish conditions for the strong uniform consistency of kernel density estimators. We also make use of this strong approximation to study the weak convergence and integrated squared error properties of these estimators. We conclude by extending our results to the setting of length-biased sampling.