Serial and nonserial sign-and-rank statistics: Asymptotic representation and asymptotic normality

• 成果类型: Article
• 署名作者: Hallin, M; Vermandele, C; Werker, B
• 署名单位: Universite Libre de Bruxelles; Universite Libre de Bruxelles; Tilburg University; Universite Libre de Bruxelles; Universite Libre de Bruxelles
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000769
• 发表日期: 2006
• 页码: 254-289
• 关键词: median regression-models; arma models; tests

摘要

The classical theory of rank-based inference is entirely based either on ordinary ranks, which do not allow for considering location (intercept) parameters, or on signed ranks, which require an assumption of symmetry. If the median, in the absence of a symmetry assumption, is considered as a location parameter, the maximal invariance property of ordinary ranks is lost to the ranks and the signs. This new maximal invariant thus suggests a new class of statistics, based on ordinary ranks and signs. An asymptotic representation theory A la Hajek is developed here for such statistics, both in the nonserial and in the serial case. The corresponding asymptotic normality results clearly show how the signs add a separate contribution to the asymptotic variance, hence, potentially, to asymptotic efficiency. As shown by Hallin and Werker [Bernoulli 9 (2003) 137-165], conditioning in an appropriate way on the maximal invariant potentially even leads to semiparametrically efficient inference. Applications to semiparametric inference in regression and time series models with median restrictions are treated in detail in an upcoming companion paper.