We introduce a new type of threshold regression models called upper hinge models. Under this type of threshold models, there only exists an association between the predictor of interest and the outcome when the predictor is less than some threshold value. Just like hinge models, upper hinge models can be seen as a special case of the more general segmented or two-phase regression models. The importance of studying upper hinge models is that even though they only have one fewer degree of freedom than segmented models, they can be estimated with much greater efficiency. We develop a new fast grid search algorithm to estimate upper hinge linear regression models. The new algorithm reduces the computational complexity of the search algorithm dramatically and renders the existing fast grid search algorithm inadmissible. The fast grid search algorithm makes it feasible to construct bootstrap confidence intervals for upper hinge linear regression models; for upper hinge generalized linear models of non-Gaussian family, we derive asymptotic normality to facilitate construction of model-robust confidence intervals. We perform numerical experiments and illustrate the proposed methods with two real data examples from the ecology literature.

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上铰链回归模型的估计与推断

我们介绍了一种称为上铰链模型的新型阈值回归模型。在这种类型的阈值模型下，只有当预测变量小于某个阈值时，感兴趣的预测变量与结果之间才会存在关联。就像铰链模型一样，上铰链模型可以看作是更一般的分段或两阶段回归模型的特例。研究上铰链模型的重要性在于，尽管它们只比分段模型少一个自由度，但它们的估计效率要高得多。我们开发了一种新的快速网格搜索算法来估计上铰链线性回归模型。新算法大大降低了搜索算法的计算复杂度，并使现有的快速网格搜索算法无法接受。快速网格搜索算法使得构建上铰链线性回归模型的自举置信区间成为可能；对于非高斯族的上铰链广义线性模型，我们推导了渐近正态性以促进模型稳健置信区间的构建。我们进行了数值实验，并用生态学文献中的两个真实数据示例说明了所提出的方法。