Multivariate Generalized Linear-statistics of short range dependent data

TL;DR

This paper establishes a central limit theorem for generalized linear statistics of dependent data, extending their applicability to strongly mixing sequences and providing practical tools like confidence intervals for tail-parameter estimation.

Contribution

It introduces a limit theorem for GL-statistics of dependent data and demonstrates their use in tail-parameter estimation with subsampling methods.

Findings

Central limit theorem for GL-statistics under strong mixing.

Effective subsampling method for confidence interval estimation.

Application to Pareto tail-parameter estimation.

Abstract

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 strong mixing in simulations.