demofit (version 0.1.4)

CBDQCS: CBD with curvature and cohort model

Description

Fits and forecasts mortality rates using CBD with curvature and cohort model.

Usage

CBDQCS(
  x,
  M,
  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 CBDQCS with associated S3 methods coef, forecast (which = 1 for smoothed (default); which = 2 for raw), plot, residuals, and simulate (nsim for setting number of simulations; seed for initialising random number generator).

Arguments

x

vector of ages.

M

matrix of mortality rates (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 with curvature and cohort (M7) model is specified as

\(ln(m_{x,t}) = \kappa_{1,t} + \kappa_{2,t} (x-\bar{x}) + \kappa_{3,t} ((x-\bar{x})^2-\sigma^2) + \gamma_{t-x} + \epsilon_{x,t}\).

The model is estimated by Newton updating scheme and is forecasted by ARIMA applied to \(\kappa_{1,t}\), \(\kappa_{2,t}\), \(\kappa_{3,t}\), and \(\gamma_c\). Constraints include sum of \(\gamma_c\) is zero, sum of \(c\gamma_c\) is zero, and sum of \(c^{2}\gamma_c\) is zero. It is designed for ages 50-90.

References

Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., and Balevich, I. (2009). A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13(1), 1-35.

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))
fit <- CBDQCS(x=x,M=M,curve="makeham",h=30,jumpoff=2)
coef(fit)
forecast::forecast(fit)
plot(fit)
residuals(fit)