Quantile regression is a popular and powerful method for studying the effect of regressors on quantiles of a response distribution. However, existing results on quantile regression were mainly developed for cases in which the quantile level is fixed, and the data are often assumed to be independent. Motivated by recent applications, we consider the situation where (i) the quantile level is not fixed and can grow with the sample size to capture the tail phenomena, and (ii) the data are no longer independent, but collected as a time series that can exhibit serial dependence in both tail and non-tail regions. To study the asymptotic theory for high-quantile regression estimators in the time series setting, we introduce a tail adversarial stability condition, which had not previously been described, and show that it leads to an interpretable and convenient framework for obtaining limit theorems for time series that exhibit serial dependence in the tail region, but are not necessarily strongly mixing. Numerical experiments are conducted to illustrate the effect of tail dependence on high-quantile regression estimators, for which simply ignoring the tail dependence may yield misleading |$p$|-values.

中文翻译：

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尾部相关时间序列的高分位数回归

分位数回归是研究回归变量对响应分布分位数的影响的一种流行而强大的方法。但是，有关分位数回归的现有结果主要是针对分位数级别固定的情况而开发的，并且通常假定数据是独立的。受最新应用程序的启发，我们考虑以下情况：（i）分位数水平不固定，并且可以随样本大小的增长而增长，以捕获尾部现象；（ii）数据不再独立，而是作为一个时间序列收集在尾巴和非尾巴区域均可表现出序列依赖性。为了研究时间序列中高分位数回归估计量的渐近理论，我们引入了一个尾部对抗性稳定条件，该条件以前没有描述过，并表明，这为获得时间序列的极限定理提供了一个可解释且方便的框架，该极限定理在尾部区域表现出序列相关性，但不一定会强烈混合。进行了数值实验，以说明尾部相关性对高分位数回归估计量的影响，为此，仅忽略尾部相关性可能会产生误导| $ p $ | 值。