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Last week, while I was giving my crash course on R for insurance, we’ve been discussing possible extensions of Lee & Carter (1992) model. If we look at the seminal paper, the model is defined as follows

Hence, it means that $\mathbb{E}[\log \mu_{x,t}] =\alpha_x+\beta_x\cdot \kappa_t$ This would be a (non)linear model on the logarithm of the mortality rate. A non-equivalent, but alternative expression might be

$\log\mathbb{E}[ \mu_{x,t}] =\alpha_x+\beta_x\cdot \kappa_t$

which could be obtained as a Gaussian model, with a log-link function $\mu_{x,t}\sim{\mathcal{N}(e^{\alpha_x+\beta_x\cdot \kappa_t},\sigma^2)$ Actually, this model can be compared to the more popular one, introduced in 2002 by Natacha Brouhns, Michel Denuit and Jeroen Vermunt, where a Poisson regression is used to count deaths (with the exposure used as an offset variable) $D_{x,t}\sim{\mathcal{P}(E_{x,t}\cdot e^{\alpha_x+\beta_x\cdot \kappa_t})$ On our datasets

EXPO <- read.table(
"http://freakonometrics.free.fr/Exposures-France.txt",
"http://freakonometrics.free.fr/Deces-France.txt",
header=TRUE,skip=0) ### !!!! 0
base=data.frame(
D=DEATH$Total, E=EXPO$Total,
X=as.factor(EXPO$Age), T=as.factor(EXPO$Year))
library(gnm)
listeage=c(101:109,"110+")
sousbase=base[! base$X %in% listeage,] # on met des nombres car il faut calculer T-X sousbase$X=as.numeric(as.character(sousbase$X)) sousbase$T=as.numeric(as.character(sousbase$T)) sousbase$C=sousbase$T-sousbase$X
sousbase$E=pmax(sousbase$E,sousbase\$D)

The codes to fit those models are the following

LC.gauss <- gnm(D/E~
as.factor(X)+
Mult(as.factor(X),as.factor(T)),
data=sousbase)

LC.gauss.2 <- gnm(log(D/E)~
as.factor(X)+
Mult(as.factor(X),as.factor(T)),
data=sousbase)

while for the Poisson regression is

LC.poisson <- gnm(D~offset(log(E))+
as.factor(X)+
Mult(as.factor(X),as.factor(T)),
data=sousbase)

To visualize the first component, the $\alpha_x$‘s, use

alphaG=coefficients(LC.gauss)[1]+c(0,
coefficients(LC.gauss)[2:101])
s=sd(residuals(LC.gauss.2))

alphaG2=coefficients(LC.gauss.2)[1]+c(0,
coefficients(LC.gauss.2)[2:101])

alphaGw=coefficients(LC.gauss.w)[1]+c(0,
coefficients(LC.gauss.w)[2:101])

We can then plot them

plot(0:100,alphaP,col="black",type="l",
xlab="Age")
lines(0:100,alphaG,col="blue")
legend(0,-1,c("Poisson","Gaussian"),
lty=1,col=c("black","blue"))

On small probabilities, the difference can be considered as substential. But for elderly, it seems that the difference is rather small. Now, the problem with a Poisson model is that it might generate a lot of deaths. Maybe more than the exposure actually. A natural idea is to consider a binomial model (which is a standard model in actuarial textbooks) $D_{x,t}\sim{\mathcal{B}\left(E_{x,t},\frac{e^{\alpha_x+\beta_x\cdot \kappa_t}}{1+e^{\alpha_x+\beta_x\cdot \kappa_t}}\right)$ The codes to run that (non)linear regression would be

LC.binomiale <- gnm(D/E~
as.factor(X)+
Mult(as.factor(X),as.factor(T)),
weights=E,
data=sousbase)

One more time, we can visualize the series of $\alpha_x$‘s.

alphaB=coefficients(LC.binomiale)[1]+c(0,
coefficients(LC.binomiale)[2:101])

Here, the difference is only on old people. For small probabilities, the binomial model can be approximated by a Poisson model. Which is what we observe. On elderly people, there is a large difference, and the Poisson model underestimates the probability of dying. Which makes sense, actually, since the number of deaths has to be smaller than the exposure. A Poisson model with a large parameter will have a (too) large variance. So the model will underestimate the probability. This is what we observe on the right. It is clearly a more realistic fit.