线性阈值是否足够好？曲率诱导阈值位移的无标度参数与充分性检验
Is the Linear Threshold Good Enough? A Scale-Free Parameter and Adequacy Test for Curvature-Induced Threshold Displacement
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中文总结 AI 辅助
针对曲线函数线性化求阈值可能产生位移的问题，提出无标度参数COT及充分性检验，以判断线性阈值是否在容差内准确，并提供效应量和诊断工具。
中文摘要 AI 辅助
应用工作通常通过将光滑函数在参考点处线性化并求解交点来定位阈值。当函数弯曲时，即使标准误差有效，线性交点也可能发生实质性位移。我们引入曲率夸大参数 \\(\Theta_{COT}=\log(|h_2^*|/|h_1^*|)\\)，即二阶与一阶阈值位移的对数比率。在与线性解连续的二次分支上，\\(|h_2^*/h_1^*|=2/(1+\sqrt{1-u})\\)，其中 \\(u=2qa/b^2\\) 是由局部间隙、斜率和曲率构成的无量纲指数。因此，\\(\Theta_{COT}\\) 是无标度的，仅通过一个标量依赖于局部参数，其正则域范围为 \\((-\infty,\log 2)\\)，在相切处边界极限为 \\(\log 2\\)。我们推导了正则渐近推断，刻画了局部到相切和弱斜率失效的情况，并给出了将二阶交点与真实阈值联系起来的余项界。主要的实际贡献是一个充分性检验，该检验可以确认线性阈值在预先指定的比例容差内是准确的，而不是将未能检测到曲率视为充分性的证据。蒙特卡洛结果证实了正则情形的校准、相切附近的预测非标准行为以及弱斜率下的失效，而自助法诊断则识别出不应使用正则推断的情形。因此，COT 提供了一个效应量尺度、充分性检验和诊断工具，用于判断一阶阈值是否足够准确以进行报告。
英文摘要
Applied work often locates a threshold by linearizing a smooth function about a reference point and solving for the crossing. When the function is curved, the linear crossing can be substantively displaced even when standard errors are valid. We introduce the curvature-overstatement parameter $Θ_{COT}=\log(|h_2^*|/|h_1^*|)$, the log ratio of second- to first-order threshold displacement. On the quadratic branch continuous with the linear solution, $|h_2^*/h_1^*|=2/(1+\sqrt{1-u})$, where $u=2qa/b^2$ is a dimensionless index formed from the local gap, slope, and curvature. Thus $Θ_{COT}$ is scale-free, depends on the local parameters through one scalar, and has regular-domain range $(-\infty,\log 2)$, with boundary limit $\log 2$ at tangency. We derive regular asymptotic inference, characterize local-to-tangency and weak-slope failures, and give a remainder bound linking the second-order crossing to the true threshold. The main practical contribution is an adequacy test that can affirm that the linear threshold is accurate within a prespecified proportional tolerance, rather than treating failure to detect curvature as evidence of adequacy. Monte Carlo results confirm regular-case calibration, the predicted nonstandard behavior near tangency, and failure under weak slope, while bootstrap diagnostics identify regimes in which regular inference should not be used. COT therefore provides an effect-size scale, adequacy test, and diagnostics for deciding whether a first-order threshold is accurate enough to report.
发表机构
- Michigan State University（密歇根州立大学）
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