U-processes, U-quantile processes and generalized linear statistics of dependent data
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.
Volume (Year): 122 (2012)
Issue (Month): 3 ()
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- Arcones, Miguel A. & Giné, Evarist, 1995. "On the law of the iterated logarithm for canonical U-statistics and processes," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 217-245, August.
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- Arcones, M. A., 1993. "The Law of the Iterated Logarithm for U-Processes," Journal of Multivariate Analysis, Elsevier, vol. 47(1), pages 139-151, October.
- Wendler, Martin, 2011. "Bahadur representation for U-quantiles of dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1064-1079, July.
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