PCFM2S: Poisson common factor model with global age pattern

In demofit: Parametric Mortality Curve Fitting and Mortality Forecasting Tools

Poisson common factor model with global age pattern

Description

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

Usage

PCFM2S(
  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
)

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 factor model with global age pattern with Poisson assumption is specified as

ln(m_{x,t,i}) = \alpha_{x,i} + B_x K_t + \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 K_t and \kappa_{t,i}. Constraints include sum of B_x is one, sum of K_t is zero, sum of \beta_x is one, and sum of \kappa_{t,i} is zero. It can be applied to whole age range.

Value

An object of class PCFM2S 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.

References

Li, J., Wang, M., Liu, J., and Leung, J.W.Y. (2026). Financial valuation of retirement village via stochastic modelling of disability prevalence rates. ASTIN Bulletin, 56(2), 447-473.

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)
K <- c(9.66,9.89,10.66,9.83,9.52,7.39,7.64,6.36,2.32,4.18,
2.91,-0.61,0.28,-0.38,-1.79,-3.34,-1.74,-3.50,-4.28,-4.77,
-4.98,-7.13,-5.09,-6.41,-5.56,-5.65,-6.12,-5.64,-7.35,-6.28)
b <- c(-0.00010,0.01195,0.03030,0.02170,0.03690,0.02365,0.02280,0.03850,0.05845,0.04415,
0.04185,0.05175,0.03670,0.04195,0.04090,0.02775,0.04990,0.02865,0.03935,0.03820,
0.04000,0.02790,0.03705,0.03370,0.02940,0.02850,0.03400,0.02310,0.02675,0.03430)
k1 <- c(-1.24,-1.38,-3.48,-2.51,-1.32,-1.90,-3.42,-0.94,0.24,-0.48,
-0.26,2.70,1.39,-0.46,1.74,2.53,0.90,1.43,0.76,2.48,
0.74,2.32,0.42,1.69,-0.64,1.30,0.19,-0.69,-1.11,-1.01)
k2 <- c(2.35,0.62,-0.38,0.12,0.00,0.80,-1.39,0.38,2.47,0.40,
0.76,3.06,1.42,-0.73,0.79,1.94,0.12,0.60,-0.43,0.29,
0.17,0.98,-1.01,-0.13,-2.46,-1.24,-1.65,-2.48,-2.32,-3.06)
set.seed(123)
M1 <- exp(outer(k1,b)+outer(K,B)+matrix(a1,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.07))
M2 <- exp(outer(k2,b)+outer(K,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 <- PCFM2S(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)

demofit documentation built on June 12, 2026, 1:07 a.m.