Mortality data over time can be arranged in triangles, and models fit to such data are similar to those used in casualty loss development–particularly those that model trends in the three dimensions of calendar year of finalization, age at finalization, and origin year. We fit such models to U.S. population male and female mortality data for death (finalization) ages 55 to 89, with several distributions of residuals. The information matrix is used to estimate parameter standard deviations.
Although there is an extensive literature on fitting these models, most of the papers do not address parameter significance through t statistics, etc. and doing so finds problems with the standard models. One problem is overparameterization, and a conclusion here is that parameter reduction methods such as smoothing should be used. Other authors have tried this, but a sticky issue is finding parameter reduction methods that actually produce improvements in goodness of fit, as measured by AIC, etc. This is an open problem as far as we know and a direction for future research.
Typically the starting point for the distribution of model residuals is Poisson, but several authors have found that negative binomial fits better. Unfortunately, some of these have misinterpreted the derivation of the negative binomial as a gamma-mixed Poisson to conclude that the negative binomial arises because there are different subpopulations each with different Poisson distributions. But a sum of subpopulations each Poisson distributed is itself Poisson distributed. The mixture becomes interesting when you are drawing at random from a subpopulation whose parameter you do not know. Probably the negative binomial arises from other contagion effects, like weather, disease outbreaks, etc. Unfortunately, these also make residuals across cells not independent, and this effect has been found in other studies as well.
A few alternative ways of parameterizing negative binomial residuals are discussed, and these are also applied to the Poisson-Inverse Gaussian distribution and its generalization, the Sichel. For females the negative binomial fits best but the male data is a bit more skewed than the negative binomial. However the Poisson inverse- aussian appears to be too skewed for this data. The Sichel is more flexible, with one more parameter, and fits best.
Further insight into the shifts in mortality over time is provided by fitting Makeham-like curves to each year of death. One implication from this exercise is that male mortality trends at the older ages had a shift in 1988, possibly data related. Probably data older than that is not reliable, or at minimum comes from a different process. The overall conclusion is that more work is needed to come up with reasonable models for mortality trend, with parameter reduction a leading candidate. For trending, ARIMA models have often been fit to the calendar-year parameters after first differencing for stability. But since the parameters are estimated with error, differencing induces an autocorrelation, so the ARIMA models could be mostly fitting this artifact. Alternatives are discussed.
Keywords: Mortality Risk; Lee-Carter Model; Cohort Effects; Parameter Risk; Model Risk
