U-processes, U-quantile processes and generalized linear statistics of dependent data
AbstractGeneralized linear statistics are a unifying class that contains U-statistics, U-quantiles, L-statistics as well as trimmed and Winsorized U-statistics. For example, many commonly used estimators of scale fall into this class. GL-statistics have only been studied under independence; in this paper, we develop an asymptotic theory for GL-statistics of sequences which are strongly mixing or L1 near epoch dependent on an absolutely regular process. For this purpose, we prove an almost sure approximation of the empiricalU-process by a Gaussian process. With the help of a generalized Bahadur representation, it follows that such a strong invariance principle also holds for the empirical U-quantile process and consequently for GL-statistics. We obtain central limit theorems and laws of the iterated logarithm for U-processes, U-quantile processes and GL-statistics as straightforward corollaries.
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Bibliographic InfoArticle provided by Elsevier in its journal Stochastic Processes and their Applications.
Volume (Year): 122 (2012)
Issue (Month): 3 ()
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Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description
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