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This paper analyzes the statistical validity of using likelihood-ratio tests to determine the necessity of a Makeham term (background mortality) in mortality models, addressing the boundary issue where the parameter cannot be negative. It demonstrates that under standard conditions, the test statistic converges to a mixture of a point mass at zero and a chi-squared distribution with one degree of freedom, requiring a critical value of 2.71 at the 5% level instead of the conventional 3.84. The study also quantifies the test's local power and highlights that this calibration fails when a gamma-frailty variance is estimated and its true value is zero, leading to persistent excess rejections.
A single boundary parameter in mortality models invalidates standard likelihood-ratio test calibration, demanding a critical value of 2.71 instead of 3.84 at the 5% level.
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.