CONVERGENCE OF LINEAR FUNCTIONALS OF THE GRENANDER ESTIMATOR UNDER MISSPECIFICATION

• 成果类型: Article
• 署名作者: Jankowski, Hanna
• 署名单位: York University - Canada
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1196
• 发表日期: 2014
• 页码: 625-653
• 关键词: maximum-likelihood-estimation; Asymptotic Normality; limit distribution; concave majorant; density; error

摘要

Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate n(1/3) if the true density is curved [Sankhya Ser. A 31 (1969) 23-36] and at rate n(1/2) if the density is flat [Ann. Probab. 11 (1983) 328-345; Canad. J. Statist. 27 (1999) 557-566]. In the case that the true density is misspecified, the results of Patilea [Ann. Statist. 29 (2001) 94-123] tell us that the global convergence rate is of order n1/3 in Hellinger distance. Here, we show that the local convergence rate is n(1/2) at a point where the density is misspecified. This is not in contradiction with the results of Patilea [Ann. Statist. 29 (2001) 94-123]: the global convergence rate simply comes from locally curved well-specified regions. Furthermore, we study global convergence under misspecification by considering linear functionals. The rate of convergence is n(1/2) and we show that the limit is made up of two independent terms: a mean-zero Gaussian term and a second term (with nonzero mean) which is present only if the density has well-specified locally flat regions.