Boundary inference for the Makeham term in mortality models
Abstract
Whether a mortality model needs a Makeham term, the constant representing background mortality, is commonly decided with a likelihood-ratio test. The constant cannot be negative, so testing its absence places it on a boundary, where the usual chi-squared calibration fails. 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: at the 5% level the critical value is 2.71, not 3.84. We state conditions under which the correction holds for Makeham models, verify them for Gompertz--Makeham and gamma-Gompertz--Makeham, and quantify the information about the constant, which fixes the local power, the smallest detectable term, and the loss of detectability as the age window narrows. When a gamma-frailty variance is estimated at a true value of zero, the limit is no longer a chi-squared mixture and the correction rejects too often. Monte Carlo experiments show the conventional cutoff rejecting at half the nominal level, a correctly calibrated test missing a term at its detection threshold in most samples, and excess rejections persisting as exposure grows.
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BibTeX
@misc{patricio2026boundary,
title = {{Boundary inference for the Makeham term in mortality models}},
author = {Silvio C. Patricio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03685},
eprint = {2610.03685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03685},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boundary inference for the Makeham term in mortality models
AU - Silvio C. Patricio
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.03685
UR - https://arxiv.org/abs/2610.03685
AB - Whether a mortality model needs a Makeham term, the constant representing background mortality, is commonly decided with a likelihood-ratio test. The constant cannot be negative, so testing its absence places it on a boundary, where the usual chi-squared calibration fails. 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: at the 5% level the critical value is 2.71, not 3.84. We state conditions under which the correction holds for Makeham models, verify them for Gompertz--Makeham and gamma-Gompertz--Makeham, and quantify the information about the constant, which fixes the local power, the smallest detectable term, and the loss of detectability as the age window narrows. When a gamma-frailty variance is estimated at a true value of zero, the limit is no longer a chi-squared mixture and the correction rejects too often. Monte Carlo experiments show the conventional cutoff rejecting at half the nominal level, a correctly calibrated test missing a term at its detection threshold in most samples, and excess rejections persisting as exposure grows.
ER -