# rh: Create a Renshaw and Haberman (Lee-Carter with cohorts)... In StMoMo: Stochastic Mortality Modelling

## 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

 1 2 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 β_x^{(0)}. It can take values: "NP" for a non-parametric age term or "1" for β_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

η_{xt} = α_x + β^{(1)}_xκ_t + β^{(0)} γ_{t-x}.

or

η_{xt} = α_x + β^{(1)}_xκ_t + γ_{t-x}.

depending on the value of argument cohortAgeFun.

To ensure identifiability the following constraints are imposed

∑_tκ_t = 0, ∑_xβ^{(1)}_x = 1, ∑_cγ_c = 0

plus

∑_xβ^{(0)}_x = 1

if cohortAgeFun = "NP"

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

∑_c (c-\bar{c})γ_c = 0

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

By default β^{(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.

fit.rh, StMoMo, lc, apc
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 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)