序列与非序列符号秩统计量：渐近表示与渐近正态性
Serial and nonserial sign-and-rank statistics: asymptotic representation and asymptotic normality
摘要：
基于秩的推理的经典理论完全基于不考虑位置（截距）参数的普通秩，或者基于需要对称性假设的符号秩。如果在没有对称性假设的情况下，将中值视为位置参数，则普通秩的最大不变性属性将丧失于秩和符号。这种新的最大不变量因此提出了一种新的统计类别，基于普通的秩和符号。本文对这类统计量，在非序列和序列情形下，发展了一个拉·哈耶克的渐近表示论。相应的渐近正态性结果清楚地表明了符号如何为渐近方差增加单独的贡献，从而潜在地为渐近有效性增加贡献。如Hallin和Werker [Bernoulli 9（2003）137- 165]所示，以适当的方式对最大不变量进行条件化甚至可能导致半参数有效推理。半参数推理在回归和时间序列模型与中位数限制的应用将在即将到来的配套文件中详细处理。
英文摘要：
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 à la Hájek 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.
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