demofit (version 0.1.4)

PCAES: Poisson common age effect model

Description

Fits and forecasts mortality rates of two populations using common age effect model with Poisson assumption.

Usage

PCAES(
  x,
  D1,
  D2,
  E1,
  E2,
  curve = c("gompertz", "makeham", "oppermann", "thiele", "wittsteinbumsted", "perks",
    "weibull", "vandermaen", "beard", "heligmanpollard", "rogersplanck", "siler",
    "martinelle", "thatcher", "gompertz2", "makeham2", "oppermann2", "thiele2",
    "wittsteinbumsted2", "perks2", "weibull2", "vandermaen2", "beard2",
    "heligmanpollard2", "rogersplanck2", "siler2", "martinelle2", "thatcher2"),
  h = 10,
  jumpoff = 1
)

Value

An object of class PCAES with associated S3 methods coef, forecast (which = 1 for smoothed (default); which = 2 for raw), plot (which = 1 gives parameter estimates (default); which = 2 gives residuals and forecasts), and residuals.

Arguments

x

vector of ages.

D1

matrix of death counts of population 1 (rows as years and columns as ages).

D2

matrix of death counts of population 2 (rows as years and columns as ages).

E1

matrix of mid-year exposures of population 1 (rows as years and columns as ages).

E2

matrix of mid-year exposures of population 2 (rows as years and columns as ages).

curve

name of mortality curve for smoothing forecasted mortality rates (including gompertz, makeham, oppermann, thiele, wittsteinbumsted, perks, weibull, vandermaen, beard, heligmanpollard, rogersplanck, siler, martinelle, thatcher, gompertz2, makeham2, oppermann2, thiele2, wittsteinbumsted2, perks2, weibull2, vandermaen2, beard2, heligmanpollard2, rogersplanck2, siler2, martinelle2, thatcher2, where first 14 curves' parameters are unconstrained and last 14 curves' parameters are generally restricted to be positive).

h

forecast horizon (default = 10).

jumpoff

if 1, forecasts are based on estimated parameters only; if 2, forecasts are anchored to observed mortality rates in final year (default = 1).

Details

The common age effect (CAE) model with Poisson assumption is specified as

\(ln(m_{x,t,i}) = \alpha_{x,i} + \beta_x \kappa_{t,i}\) and \(D_{x,t,i} ~ Poisson(E_{x,t,i} m_{x,t,i})\).

The model is estimated by Newton updating scheme and is forecasted by ARIMA applied to \(\kappa_{t,i}\). Constraints include sum of \(\beta_x\) is one and sum of \(\kappa_{t,i}\) is zero. It can be applied to whole age range.

References

Kleinow, T. (2015). A common age effect model for the mortality of multiple populations. Insurance: Mathematics and Economics, 63(C), 147-152.

Examples

x <- 60:89
a1 <- c(-5.18,-5.12,-4.98,-4.92,-4.82,-4.73,-4.66,-4.53,-4.45,-4.35,
-4.26,-4.17,-4.05,-3.95,-3.84,-3.73,-3.65,-3.52,-3.40,-3.29,
-3.14,-3.02,-2.88,-2.76,-2.64,-2.49,-2.37,-2.25,-2.12,-2.00)
a2 <- c(-4.78,-4.68,-4.57,-4.49,-4.39,-4.29,-4.19,-4.10,-4.00,-3.89,
-3.80,-3.69,-3.60,-3.49,-3.39,-3.29,-3.17,-3.07,-2.96,-2.85,
-2.71,-2.62,-2.49,-2.37,-2.26,-2.14,-2.04,-1.91,-1.82,-1.72)
b <- c(0.0381,0.0340,0.0420,0.0389,0.0423,0.0414,0.0406,0.0393,0.0415,0.0400,
0.0411,0.0362,0.0387,0.0381,0.0384,0.0385,0.0356,0.0314,0.0317,0.0337,
0.0316,0.0298,0.0284,0.0270,0.0248,0.0262,0.0205,0.0215,0.0142,0.0145)
k1 <- c(8.68,8.34,7.99,6.87,8.18,5.73,4.83,5.20,2.74,3.22,
2.99,1.59,1.67,-0.65,-0.39,-1.07,-0.95,-2.78,-3.46,-2.45,
-4.12,-4.66,-4.98,-4.58,-6.30,-4.39,-5.56,-6.52,-8.26,-6.92)
k2 <- c(11.81,11.01,10.59,10.40,9.75,8.15,6.07,6.45,4.60,4.57,
4.15,1.49,1.77,-1.08,-1.44,-0.96,-1.66,-2.25,-4.67,-4.62,
-4.38,-6.37,-6.27,-6.91,-8.22,-7.35,-8.39,-7.87,-9.72,-8.65)
set.seed(123)
M1 <- exp(outer(k1,b)+matrix(a1,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.07))
M2 <- exp(outer(k2,b)+matrix(a2,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.07))
E1 <- matrix(c(107788,108036,107481,106552,104608,100104,95803,91345,84980,79557,
75146,70559,65972,60898,55623,50522,47430,45895,41443,34774,
30531,27754,25105,22271,19437,16888,14458,12146,10038,7994),30,30,byrow=TRUE)
E2 <- E1
D1 <- round(E1*M1)
D2 <- round(E2*M2)
fit <- PCAES(x=x,D1=D1,D2=D2,E1=E1,E2=E2,curve="makeham",h=30,jumpoff=2)
coef(fit)
forecast::forecast(fit)
plot(fit)
residuals(fit)