PRHS: Poisson Renshaw-Haberman model

In demofit: Parametric Mortality Curve Fitting and Mortality Forecasting Tools

Poisson Renshaw-Haberman model

Description

Fits and forecasts mortality rates using Renshaw-Haberman model with Poisson assumption.

Usage

PRHS(
  x,
  D,
  E,
  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.
D	matrix of death counts (rows as years and columns as ages).
E	matrix of mid-year exposures (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 Renshaw-Haberman (RH) model with Poisson assumption is specified as

ln(m_{x,t}) = \alpha_x + \beta_x \kappa_t + \gamma_{t-x} and D_{x,t} ~ Poisson(E_{x,t} m_{x,t}).

The model is estimated by Newton updating scheme and is forecasted by ARIMA applied to \kappa_t and \gamma_c. Constraints include sum of \beta_x is one, sum of \kappa_t is zero, and sum of \gamma_c is zero. It can be applied to whole age range.

Value

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

References

Haberman, S. and Renshaw, A. (2011). A comparative study of parametric mortality projection models. Insurance: Mathematics and Economics, 48(1), 35-55.

Examples

x <- 60:89
a <- c(-4.8499,-4.7676,-4.6719,-4.5722,-4.4847,-4.3841,-4.2813,-4.1863,-4.0861,-3.9962,
-3.8885,-3.7896,-3.6853,-3.5737,-3.4728,-3.3718,-3.2586,-3.1474,-3.0371,-2.9206,
-2.7998,-2.6845,-2.5653,-2.4581,-2.3367,-2.2159,-2.1017,-1.9941,-1.8821, -1.7697)
b <- c(0.0283,0.0321,0.0335,0.0336,0.0341,0.0358,0.0368,0.0403,0.0392,0.0395,
0.0396,0.0399,0.0397,0.0386,0.039,0.0375,0.0367,0.0368,0.035,0.0354,
0.0336,0.0323,0.0313,0.0295,0.0282,0.0265,0.024,0.0226,0.0219,0.0183)
k <- c(12.11,10.69,11.18,9.64,9.35,8.21,6.89,5.74,4.56,3.6,
3.27,2.04,1.11,-0.44,-1.05,-1.03,-1.84,-2.9,-4.03,-4.12,
-5.18,-5.64,-6,-6.51,-6.91,-6.9,-8.32,-8.53,-9.69,-9.31)
set.seed(123)
M <- exp(outer(k,b)+matrix(a,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.035))
E <- 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)
D <- round(E*M)
fit <- PRHS(x=x,D=D,E=E,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.