DIFFERENTIABILITY OF t-FUNCTIONALS OF LOCATION AND SCATTER

• 成果类型: Article
• 署名作者: Dudley, R. M.; Sidenk, Sergiy; Wang, Zuoqin
• 署名单位: Massachusetts Institute of Technology (MIT); Johns Hopkins University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS592
• 发表日期: 2009
• 页码: 939-960
• 关键词: descriptive statistics; multivariate location; nonparametric models; cauchy distribution; donsker classes; em algorithm; likelihood; unimodality

摘要

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector mu and scatter matrix Sigma of an elliptically symmetric t distribution off R-d with degrees of freedom nu > 1 extends to an M-functional defined off all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P Putting not too much mass in hyperplanes of dimension < d, as shown for empirical measures by Kent and Tyler [Ann. Statist. 19 (1991) 2102-2119]. It will be seen here that (mu, Sigma) is analytic on U for the bounded Lipschitz norm, of for d = 1 for the sup norm on distribution functions. For k = 1, 2,..., and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (mu, Sigma) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (mu(n), Sigma(n)). In dimension d = 1 only, the t(nu) functional (mu, sigma) extends to be defined and weakly continuous at all P.