Does a mortality schedule need a Makeham term? Calibrating the likelihood-ratio test

Abstract

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.