U-quantile processes and generalized linear statistics of dependent data
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Date
2010-10-12
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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 only have been studied under independence; in this paper, we establish
the central limit theorem (CLT) and the law of the iterated logarithm
(LIL) for GL-statistics of sequences which are strongly mixing or L^1 near
epoch dependent on an absolutely regular process. We first investigate the
empirical U-process. With the help of a generalized Bahadur representation,
the CLT and the LIL for the empirical U-quantile process follow. As GL-statistics are linear functionals of the U-quantile process, the CLT and the
LIL for GL-statistics are straightforward corollaries.
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Keywords
Bahadur representation, L-Statistics, mixing, near epoch dependence, U-quantile, U-statistics