20 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
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
Studentized U-quantile processes under dependence with applications to change-point analysis
Many popular robust estimators are -quantiles, most notably the
Hodges-Lehmann location estimator and the scale estimator. We prove a
functional central limit theorem for the sequential -quantile process
without any moment assumptions and under weak short-range dependence
conditions. We further devise an estimator for the long-run variance and show
its consistency, from which the convergence of the studentized version of the
sequential -quantile process to a standard Brownian motion follows. This
result can be used to construct CUSUM-type change-point tests based on
-quantiles, which do not rely on bootstrapping procedures. We demonstrate
this approach in detail at the example of the Hodges-Lehmann estimator for
robustly detecting changes in the central location. A simulation study confirms
the very good robustness and efficiency properties of the test. Two real-life
data sets are analyzed
Noncentral limit theorem and the bootstrap for quantiles of dependent data
We will show under minimal conditions on differentiability and
dependence that the central limit theorem for quantiles holds and that the
block bootstrap is weakly consistent. Under slightly stronger conditions, the
bootstrap is strongly consistent. Without the differentiability condition, quantiles might have a non-normal asymptotic distribution and the bootstrap might
fail
Asymptotics of the two-stage spatial sign correlation
Acknowledgments This research was supported in part by the Collaborative Research Grant 823 of the German Research Foundation. The authors wish to thank the editors and referees for their careful handling of the manuscript. They further acknowledge the anonymous referees of the article Spatial sign correlation (J. Multivariate Anal. 135, pages 89–105, 2015), who independently of each other suggested to further explore the properties of two-stage spatial sign correlation.Non peer reviewedPreprin
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