demofit (version 0.1.4)

PCBDS: Poisson CBD model

Description

Fits and forecasts mortality rates using CBD model with Poisson assumption.

Usage

PCBDS(
  x,
  D,
  E,
  curve = c("gompertz", "makeham", "perks", "weibull", "beard", "martinelle", "thatcher",
    "gompertz2", "makeham2", "perks2", "weibull2", "beard2", "martinelle2", "thatcher2"),
  h = 10,
  jumpoff = 1
)

Value

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

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, perks, weibull, beard, martinelle, thatcher, gompertz2, makeham2, perks2, weibull2, beard2, martinelle2, thatcher2, where first 7 curves' parameters are unconstrained and last 7 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 CBD (M5) model with Poisson assumption is specified as

\(ln(m_{x,t}) = \kappa_{1,t} + \kappa_{2,t} (x-\bar{x})\) 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 \(\kappa_{1,t}\) and \(\kappa_{2,t}\). It is designed for ages 50-90.

References

Cairns, A.J.G., Blake, D., and Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73(4), 687-718.

Examples

x <- 60:89
k1 <- -2.97-0.0245*(0:29)
k2 <- 0.101+0.000345*(0:29)
set.seed(123)
M <- exp(matrix(k1,nrow=30,ncol=30,byrow=FALSE)+outer(k2,(x-mean(x)))+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 <- PCBDS(x=x,D=D,E=E,curve="makeham",h=30,jumpoff=2)
coef(fit)
forecast::forecast(fit)
plot(fit)
residuals(fit)