647 research outputs found
Bahadur Representation for U-Quantiles of Dependent Data
U-quantiles are applied in robust statistics, like the Hodges-Lehmann
estimator of location for example. They have been analyzed in the case of
independent random variables with the help of a generalized Bahadur
representation. Our main aim is to extend these results to U-quantiles of
strongly mixing random variables and functionals of absolutely regular
sequences. We obtain the central limit theorem and the law of the iterated
logarithm for U-quantiles as straightforward corollaries. Furthermore, we
improve the existing result for sample quantiles of mixing data
U-Processes, U-Quantile Processes and Generalized Linear Statistics of Dependent Data
Generalized linear statistics are an 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 develop an asymptotic theory for GL-statistics of sequences which are
strongly mixing or L^1 near epoch dependent on an absolutely regular process.
For this purpose, we prove an almost sure approximation of the empirical
U-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.Comment: 24 page
Law of the Iterated Logarithm for U-Statistics of Weakly Dependent Observations
The law of the iterated logarithm for partial sums of weakly dependent
processes was intensively studied by Walter Philipp in the late 1960s and
1970s. In this paper, we aim to extend these results to nondegenerate
U-statistics of data that are strongly mixing or functionals of an absolutely
regular process.Comment: typos corrrecte
Convergence of U-statistics indexed by a random walk to stochastic integrals of a Levy sheet
We establish limit theorems for U-statistics indexed by a random walk on Z^d
and we express the limit in terms of some Levy sheet Z(s,t). Under some
hypotheses, we prove that the limit process is Z(t,t) if the random walk is
transient or null-recurrent ant that it is some stochastic integral with
respect to Z when the walk is positive recurrent. We compare our results with
results for random walks in random scenery.Comment: 38 page
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