This study investigates simultaneous confidence bands for extremal quantile regressions using the spline method. We construct the spline estimator for intermediate order quantiles using a conventional quantile regression framework, and we obtain the extreme order quantile estimator by extrapolating the spline estimator for intermediate order quantiles. We establish the asymptotic normality of the spline and extrapolated estimators for intermediate and extreme order quantiles. By applying the volume of tube formula to the above two estimators, we construct simultaneous conditional quantile confidence bands for intermediate and extreme order quantiles. To confirm the performance of the proposed confidence bands, we use a Monte Carlo simulation and an example with real data.

中文翻译：

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用样条曲线进行极值分位数回归的同时置信带

这项研究使用样条方法研究了极值分位数回归的同时置信带。我们使用常规分位数回归框架构造中阶分位数的样条估计量，并通过外推中阶分位数的样条估计量来获得极阶分位数估计量。我们建立了样条和渐进估计量的渐近正态性，用于中阶和极阶分位数。通过将管公式的体积应用于上述两个估计量，我们为中阶和极阶分位数构造了同时条件分位数置信带。为了确认建议的置信带的性能，我们使用了蒙特卡洛模拟和带有实际数据的示例。