U-processes, U-quantile processes and generalized linear statistics of dependent data

Martin Wendler

Stochastic Processes and their Applications, 2012, vol. 122, issue 3, 787-807

Abstract: Generalized 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.

Keywords: L-Statistic; U-statistics; Invariance principle; Bahadur representation; Mixing; Near epoch dependence (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2011.11.010

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