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Last course will be uploaded soon (the links will be here and there). The R code considered is given below. First, we had to work a little bit on the datasets,

tabB=read.table("http://perso.univ-rennes1.fr/arthur.charpentier/tabB.csv",
ANNEE=tabB[,1]
BASEB=tabB[,seq(2,246,by=2)]
BASEB=as.matrix(BASEB[,1:100])
AGE=0:ncol(BASEB)
an=tabC[,1]
an=an[-c(16,23,43,48,51,53,106)]
BASEC=tabC[,2:101]
BASEC=as.matrix(BASEC[-c(16,23,43,48,51,53,106),])
BASEB=BASEB[,1:90]
BASEC=BASEC[,1:90]
AGE=AGE[1:90]
MU=as.matrix(log(BASEB/BASEC))
col="green")
library(rgl)
persp3d(ANNEE,AGE,MU,col="light blue")
(this last line is here to play a little bit with the 3d mortality surface). We first used the Lee-Carter function proposed by JPMorgan in LifeMetrics,
source("http://perso.univ-rennes1.fr/arthur.charpentier/fitModels.r")
x=AGE
y=ANNEE
d=BASEB
e=BASEC
w=matrix(1,nrow(d),ncol(e))
LC1=fit701(x,y,e,d,w)
plot(AGE,LC1$beta1) plot(ANNEE,LC1$kappa2)
plot(AGE,LC1$beta2) Then we considered nonlinear Poisson regression, D=as.vector(BASEB) E=as.vector(BASEC) A=rep(AGE,each=length(ANNEE)) Y=rep(ANNEE,length(AGE)) base=data.frame(D,E,A,Y,a=as.factor(A), y=as.factor(Y)) LC2=gnm(D~a+Mult(a,y),offset=log(E), family=poisson,data=base) plot(AGE[-1],LC2$coefficients[1]+LC2$coefficients[2:90]) lines(AGE,LC1$beta1,col="blue")
plot(ANNEE,LC2$coefficients[181:279]) plot(ANNEE,LC1$kappa2,col="red")
plot(AGE,LC1$beta2) As mentioned during the course, this technique is great… but it is sentive to initial values in the optimization procedure. For instance, consider the following loops, plot(AGE[-1],LC2$coefficients[1]+LC2$coefficients[2:90], type="l",col="blue",xlab="",ylab="") for(s in 1:200){ LC2=gnm(D~a+Mult(a,y),offset=log(E), family=poisson,data=base) lines(AGE[-1],LC2$coefficients[1]+LC2$coefficients[2:90],col="blue") } Here are representation of the first component in the Lee-Carter model, Hence, the estimation depends on the choice of initial values… even if the shape remains unchanged… Then, we finnished with Rob Hyndman’s package, library(demography) base=demogdata(data=t(exp(MU)),pop=t(BASEC), ages=AGE,years=ANNEE,type="mortality", label="France",name="Total",lambda=0) LC3=lca(base) LC3F=forecast(LC3,100) plot(LC3$year,LC3$kt,xlim=c(1900,2100),ylim=c(-300,150)) lines(LC3F$year,LC3$kt[99]+LC3F$kt.f$mean) lines(LC3F$year,LC3$kt[99]+LC3F$kt.f$lower,lty=2) lines(LC3F$year,LC3$kt[99]+LC3F$kt.f$upper,lty=2) We concluded with a short discussion about errors of the Lee-Carter model (on French mortality) D=as.vector(BASEB) E=as.vector(BASEC) A=rep(AGE,each=length(ANNEE)) Y=rep(ANNEE,length(AGE)) base=data.frame(D,E,A,Y,a=as.factor(A), y=as.factor(Y)) RES=residuals(LC2,"pearson") base$res=RES
plot(base$A,base$res)
couleur=heat.colors(100)
plot(base$A,base$res,col=couleur[base$Y-1898]) plot(base$Y,base$res,col=couleur[base$A+1])
The graphs can be seen below (as a function of time, and a function of ages)
his Wednesday, we will discuss how we can use those estimators (or other ones) in life insurance…