WHEN UNIFORM WEAK CONVERGENCE FAILS: EMPIRICAL PROCESSES FOR DEPENDENCE FUNCTIONS AND RESIDUALS VIA EPI- AND HYPOGRAPHS 1
WHEN UNIFORM WEAK CONVERGENCE FAILS: EMPIRICAL PROCESSES FOR DEPENDENCE FUNCTIONS AND RESIDUALS VIA EPI- AND HYPOGRAPHS 1
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当一致弱收敛失败时：通过 EPI 和 Hypographs 求依赖函数和残差的经验过程 1
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
S. Volgushev
中科院分区：
文献类型：
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作者：
Axel Bücher;J. Segers;S. Volgushev
在过去的几十年里，随机过程的弱收敛理论已成为分析各种统计渐近性质的标准工具。通常，在具有最高度量的有界函数空间中考虑弱收敛。然而，在某些情况下，这些空间中的弱收敛无法成立。示例包括线性回归模型中的经验连接和尾部依赖过程以及残差经验过程，以防基础分布缺乏一定程度的平滑度。为了解决这个问题，引入了局部有界函数的新度量，并发展了相应的弱收敛理论。新度量的收敛与外收敛和低收敛相关，并且弱于一致收敛。尽管如此，对于连续极限，它相当于局部一致收敛，而在温和的边条件下，它意味着 L p 收敛。对于上述示例，在相对于最高距离没有发生弱收敛的情况下，建立了相对于新度量的弱收敛。结果用于获得重采样过程和拟合优度检验的渐近特性。
In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain de-gree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies L p convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.