Two‐phase polynomial regression models (Robison, 1964; Fuller, 1969; Gallant and Fuller, 1973; Zhan et al., 1996) are widely used in ecology, public health, and other applied fields to model nonlinear relationships. These models are characterized by the presence of threshold parameters, across which the mean functions are allowed to change. That the threshold is a parameter of the model to be estimated from the data is an essential feature of two‐phase models. It distinguishes them, and more generally, multiphase models, from the spline models and has profound implications for both computation and inference for the models. Estimation of two‐phase polynomial regression models is a nonconvex, nonsmooth optimization problem. Grid search provides high‐quality solutions to the estimation problem, but is very slow when performed by brute force. Building upon our previous work on piecewise linear two‐phase regression models estimation, we develop fast grid search algorithms for two‐phase polynomial regression models and demonstrate their performance. Furthermore, we develop bootstrap‐based pointwise and simultaneous confidence bands for mean functions. Monte Carlo studies are conducted to demonstrate the computational and statistical properties of the proposed methods. Three real datasets are used to help illustrate the application of two‐phase models, with special attention on model choice.

中文翻译：

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连续两相多项式回归模型的快速网格搜索和基于引导程序的推理


两相多项式回归模型（Robison，1964；Fuller，1969；Gallant 和 Fuller，1973；Zhan 等，1996）广泛应用于生态学、公共卫生和其他应用领域来建模非线性关系。这些模型的特点是存在阈值参数，平均函数允许在阈值参数上发生变化。阈值是根据数据估计的模型参数，这是两相模型的一个基本特征。它将它们（更一般地说，多相模型）与样条模型区分开来，并且对模型的计算和推理具有深远的影响。两相多项式回归模型的估计是一个非凸、非光滑优化问题。网格搜索为估计问题提供了高质量的解决方案，但通过强力执行时速度非常慢。基于我们之前关于分段线性两相回归模型估计的工作，我们开发了用于两相多项式回归模型的快速网格搜索算法并展示了它们的性能。此外，我们为均值函数开发了基于引导的逐点和同步置信带。进行蒙特卡罗研究是为了证明所提出方法的计算和统计特性。使用三个真实数据集来帮助说明两阶段模型的应用，特别注意模型的选择。