rh {StMoMo}

Create a Renshaw and Haberman (Lee-Carter with cohorts) mortality model

Description

Utility function to initialise a StMoMo object representing a Renshaw and Haberman (Lee-Carter with cohorts) mortality model introduced in Renshaw and Haberman (2006).

Usage

rh(link = c("log", "logit"), cohortAgeFun = c("1", "NP"),
  approxConst = FALSE)

Arguments

link

defines the link function and random component associated with the mortality model. "log" would assume that deaths follow a Poisson distribution and use a log link while "logit" would assume that deaths follow a Binomial distribution and a logit link.

cohortAgeFun

defines the cohort age modulating parameter \beta_x^{(0)}. It can take values: "NP" for a non-parametric age term or "1" for \beta_x^{(0)}=1 (the default).

approxConst

defines if the approximate identifiability constraint of Hunt and Villegas (2015) is applied or not. If TRUE, the output object is of class rh and subsequent model fitting is performed with fit.rh. If FALSE, the output object is of class StMoMo and subsequent model fitting is performed with fit.StMoMo.

Details

The created model is either a log-Poisson or a logit-Binomial version of the Renshaw and Haberman model which has predictor structure

\eta_{xt} = \alpha_x + \beta^{(1)}_x\kappa_t + \beta^{(0)} \gamma_{t-x}.

or

\eta_{xt} = \alpha_x + \beta^{(1)}_x\kappa_t + \gamma_{t-x}.

depending on the value of argument cohortAgeFun.

To ensure identifiability the following constraints are imposed

\sum_t\kappa_t = 0, \sum_x\beta^{(1)}_x = 1, \sum_c\gamma_c = 0

plus

\sum_x\beta^{(0)}_x = 1

if cohortAgeFun = "NP"

In addition, if approxConst=TRUE then the approximate identifiability constraint

\sum_c (c-\bar{c})\gamma_c = 0

is applied to improve the stability and robustness of the model (see Hunt and Villegas (2015)).

By default \beta^{(0)}_x = 1 as this model has shown to be more stable (see Haberman and Renshaw (2011) and Hunt and Villegas (2015)).

Value

An object of class "StMoMo" or "rh".

References

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

Hunt, A., & Villegas, A. M. (2015). Robustness and convergence in the Lee-Carter model with cohorts. Insurance: Mathematics and Economics, 64, 186-202.

Renshaw, A. E., & Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556-570.

See Also

fit.rh, StMoMo, lc, apc

Examples

LCfit <-  fit(lc(), data = EWMaleData, ages.fit = 55:89)
wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
RHfit <- fit(rh(), data = EWMaleData, ages.fit = 55:89, wxt = wxt, 
             start.ax = LCfit$ax, start.bx = LCfit$bx, start.kt = LCfit$kt)
plot(RHfit)

#Impose approximate constraint as in Hunt and Villegas (2015)    
## Not run: 
RHapprox <- rh(approxConst = TRUE)
RHapproxfit <- fit(RHapprox, data = EWMaleData, ages.fit = 55:89, 
                    wxt = wxt)
plot(RHapproxfit) 

## End(Not run)