Quantile regression (QR) has gained popularity during the last decades, and is now considered a standard method by applied statisticians and practitioners in various fields. In this work, we applied QR to investigate climate change by analysing historical temperatures in the Arctic Circle. This approach proved very flexible and allowed to investigate the tails of the distribution, that correspond to extreme events. The presence of quantile crossing, however, prevented using the fitted model for prediction and extrapolation. In search of a possible solution, we first considered a different version of QR, in which the QR coefficients were described by parametric functions. This alleviated the crossing problem, but did not eliminate it completely. Finally, we exploited the imposed parametric structure to formulate a constrained optimization algorithm that enforced monotonicity. The proposed example showed how the relatively unexplored field of parametric quantile functions could offer new solutions to the long-standing problem of quantile crossing. Our approach is particularly convenient in situations, like the analysis of time series, in which the fitted model may be used to predict extreme quantiles or to perform extrapolation. The described estimator has been implemented in the R package qrcm.

分位数回归 (QR) 在过去几十年中越来越受欢迎，现在被各个领域的应用统计学家和从业者认为是一种标准方法。在这项工作中，我们通过分析北极圈的历史温度，应用 QR 来调查气候变化。事实证明，这种方法非常灵活，可以调查与极端事件相对应的分布尾部。然而，分位数交叉的存在阻止了使用拟合模型进行预测和外推。为了寻找可能的解决方案，我们首先考虑了不同版本的 QR，其中 QR 系数由参数描述职能。这减轻了交叉问题，但并没有完全消除它。最后，我们利用强加的参数结构来制定强制单调性的约束优化算法。提议的例子展示了参数分位数函数这个相对未开发的领域如何为长期存在的分位数交叉问题提供新的解决方案。我们的方法在诸如时间序列分析之类的情况下特别方便，在这种情况下，拟合模型可用于预测极端分位数或执行外推。所描述的估计器已在 R 包 qrcm 中实现。