Generalized linear statistics for near epoch dependent processes with application to EGARCH-processes

Abstract: The class of Generalized $L$-statistics ($GL$-statistics) unifies a broad class of different estimators, for example scale estimators based on multivariate kernels. $GL$-statistics are functionals of $U$-quantiles and therefore the dimension of the kernel of the $U$-quantiles determines the kernel dimension of the estimator. Up to now only few results for multivariate kernels are known. Additionally, most theory was established under independence or for short range dependent processes. In this paper we establish a central limit theorem for $GL$-statistics of functionals of short range dependent data, in particular near epoch dependent sequences on absolutely regular processes, and arbitrary dimension of the underlying kernel. This limit theorem is based on the theory of $U$-statistics and $U$-processes, for which we show a central limit theorem as well as an invariance principle. The use of near epoch dependent processes admits us to consider functionals of short range dependent processes and therefore models like the EGARCH-model. We also develop a consistent estimator of the asymptotic variance of $GL$-statistics.