发表机构
Interdisciplinary Center on Population Dynamics; University of Southern Denmark（人口动力学跨学科中心; 南丹麦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对死亡率模型是否需要Makeham项，提出边界参数下似然比检验的校正方法，将临界值从3.84调整为2.71，并验证其适用条件及功效特性。
AI 中文摘要
拟合的死亡率模型是否需要 Makeham 项（表示背景死亡率的非负常数），通常通过似然比检验来决定。由于该常数不能为负，检验其不存在时将参数置于其取值范围的边界上，因此通常的卡方校准不再适用。假设独立泊松死亡计数和标准正则条件，我们证明当该常数是唯一处于边界上的参数时，统计量收敛于零处点质量与自由度为1的卡方分布的等量混合，因此在5%显著性水平下临界值为2.71而非3.84。我们给出了校正对Makeham模型成立的条件，对Gompertz-Makeham和gamma-Gompertz-Makeham模型进行了验证，并量化了数据关于该常数所携带的信息，这决定了检验的局部功效、其能检测到的最小项，以及随着年龄窗口变窄可检测性下降的速度。当估计gamma脆弱性方差且其真实值也为零时，极限不再是卡方混合，通常的校正会过度拒绝。蒙特卡洛实验表明，传统临界值在名义水平的一半处拒绝，正确校准的检验在大多数样本中仍可能在其自身检测阈值处遗漏一项，并且当估计的零脆弱性方差存在时，随着暴露量的增加，过度拒绝现象持续存在。
英文摘要
Whether a fitted mortality model needs a Makeham term, the non-negative constant representing background mortality, is commonly decided with a likelihood-ratio test. Because the constant cannot be negative, testing its absence places the parameter on the boundary of its range, and the usual chi-squared calibration does not apply. Assuming independent Poisson death counts and standard regularity conditions, we show that when the constant is the only parameter on a boundary the statistic converges to an equal mixture of a point mass at zero and a chi-squared distribution with one degree of freedom, so that at the 5% level the critical value is 2.71 rather than 3.84. We give conditions under which the correction holds for Makeham models, verify them for Gompertz-Makeham and gamma-Gompertz-Makeham, and quantify the information the data carry about the constant, which fixes the local power of the test, the smallest term it can detect, and how fast detectability falls as the age window narrows. When a gamma-frailty variance is estimated and its true value is also zero, the limit is no longer a chi-squared mixture and the usual correction rejects too often. Monte Carlo experiments show that the conventional cutoff rejects at half the nominal level, that a correctly calibrated test can still miss a term at its own detection threshold in most samples, and that excess rejections under an estimated zero frailty variance persist as exposure grows.
Comments16 pages, 2 figures, 1 table. Code and reproducibility materials: https://github.com/scpatricio/Makeham_LRT
