近历元相关过程的广义线性统计量及其在EGARCH过程中的应用
Generalized linear statistics for near epoch dependent processes with application to EGARCH-processes
摘要：
广义$L$ -统计量（$GL$ -统计量）类统一了广泛的不同估计量，例如基于多元核的规模估计量。$GL$ -统计量是$U$ -分位数的泛函，因此$U$ -分位数的核的维数确定估计量的核维数。到目前为止，只有少数结果的多变量内核是已知的。此外，大多数理论都是在独立或短程依赖过程下建立的。本文建立了一个中心极限定理，$GL$ -统计量的泛函的短程相关数据，特别是近历元相关序列的绝对正则过程，和任意维数的核心。这个极限定理是建立在$U$ -统计量和$U$ -过程理论的基础上的，我们证明了它的一个中心极限定理和一个不变原理.使用近时代的依赖过程允许我们考虑泛函的短程依赖过程，因此像EGARCH模型的模型。我们还开发了一个一致的估计的渐近方差的$GL$ -统计量。
英文摘要：
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.
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