Electronic Journal of Statistics

Multivariate generalized linear-statistics of short range dependent data

Svenja Fischer, Roland Fried, and Martin Wendler

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Generalized linear ($GL$-) statistics are defined as functionals of an $U$-quantile process and unify different classes of statistics such as $U$-statistics and $L$-statistics. We derive a central limit theorem for $GL$-statistics of strongly mixing sequences and arbitrary dimension of the underlying kernel. For this purpose we establish a limit theorem for $U$-statistics and an invariance principle for $U$-processes together with a convergence rate for the remaining term of the Bahadur representation.

An application is given by the generalized median estimator for the tail-parameter of the Pareto distribution, which is commonly used to model exceedances of high thresholds. We use subsampling to calculate confidence intervals and investigate its behaviour under independence and under strong mixing in simulations.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 646-682.

Received: December 2014
First available in Project Euclid: 7 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions 60G10: Stationary processes
Secondary: 60F17: Functional limit theorems; invariance principles

$GL$-statistics $U$-statistics strong mixing generalized median estimator tail parameter


Fischer, Svenja; Fried, Roland; Wendler, Martin. Multivariate generalized linear-statistics of short range dependent data. Electron. J. Statist. 10 (2016), no. 1, 646--682. doi:10.1214/16-EJS1124. https://projecteuclid.org/euclid.ejs/1457382317

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