Package 'StanMoMo'

The 'StanMoMo' package.

Description

The StanMoMo package performs Bayesian Mortality Modeling with Stan for a variety of popular mortality models. The current package supports the Lee-Carter model, the Renshaw-Haberman model, the Age-Period-Cohort model, the Cairns-Blake-Dowd model and the M6 model. By a simple call, the user inputs deaths and exposures and the package outputs the MCMC simulations for each parameter, the log likelihoods and predictions. Moreover, the package includes tools for model selection and Bayesian model averaging by leave-future-out validation.

References

Stan Development Team (2020). RStan: the R interface to Stan. R package version 2.19.3. https://mc-stan.org

---

Bayesian Age-Period-Cohort model with 'Stan'

Description

Fit and Forecast Bayesian APC model. The model can be fitted with a Poisson or Negative-Binomial distribution. The function outputs posteriors distributions for each parameter, predicted death rates and log-likelihoods.

Usage

apc_stan(
  death,
  exposure,
  forecast,
  validation = 0,
  family = c("poisson", "nb"),
  ...
)

Arguments

death	Matrix of deaths.
exposure	Matrix of exposures.
forecast	Number of years to forecast.
validation	Number of years for validation.
family	specifies the random component of the mortality model. "Poisson" assumes a Poisson model with log link and "nb" assumes a negative-binomial model with log link and overdispersion parameter $\phi$.
...	Arguments passed to rstan::sampling (e.g. iter, chains).

Details

The created model is either a log-Poisson or a log-Negative-Binomial version of the APC model:

$D_{x,t} \sim \mathcal{P}(\mu_{x,t} e_{x,t})$

or

$D_{x,t}\sim NB\left(\mu_{x,t} e_{x,t},\phi\right)$

with

$\log \mu_{xt} = \alpha_x + \kappa_t + \gamma_{t-x}.$

To ensure the identifiability of th model, we impose

$\kappa_1=0, \gamma_1=0,\gamma_C=0,$

where $C$ represents the most recent cohort in the data.

For the priors, we assume that

$\alpha_x \sim N(0,100),\frac{1}{\phi} \sim Half-N(0,1).$

For the period term, similar to the LC model, we consider a random walk with drift:

$\kappa_{t}=c+\kappa_{t-1}+\epsilon_{t},\epsilon_{t}\sim N(0,\sigma^2)$

with the following hyperparameters assumptions: $c \sim N(0,10),\sigma \sim Exp(0.1)$.

For the cohort term, we consider a second order autoregressive process (AR(2)):

$\gamma_{c}=\psi_1 \gamma_{c-1}+\psi_2 \gamma_{c-2}+\epsilon^{\gamma}_{t},\quad \epsilon^{\gamma}_{t}\sim N(0,\sigma_{\gamma}).$

To close the model specification, we impose some vague priors assumptions on the hyperparameters:

$\psi_1,\psi_2 \sim N(0,10),\quad \sigma_{\gamma}\sim Exp(0.1).$

Value

An object of class stanfit returned by rstan::sampling

References

Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., Ong, A., & 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

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-Poisson model
ages.fit<-70:90
years.fit<-1990:2010
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-50 # Toy example, consider at least 2000 iterations
fitAPC=apc_stan(death = deathFR,exposure=exposureFR, forecast = 5, family = "poisson",
iter=iterations,chains=1)

---

Boxplot for the posterior distribution

Description

The function generates boxplots for the posterior distribution of the main parameters

Usage

boxplot_post_dist(stan_fit, parm_name, ages, years)

Arguments

stan_fit	stan fit object
parm_name	string to indicate the name of the parameter, to choose from c('a', 'b', 'g', 'k', 'k2', 'phi')
ages	range of ages
years	range of years

Value

Posterior distribution shown as boxplots

Examples

years <- 1990:2017
ages <- 50:90
cohorts <- sort(unique(as.vector(sapply(years, function(year) year - ages))))
death <- FRMaleData$Dxt[formatC(ages),formatC(years)]
exposure <- FRMaleData$Ext[formatC(ages),formatC(years)]
iterations<-50 # Toy example, consider at least 2000 iterations
stan_fit <- fit_mo_mo("m6", death , exposure, ages, 0, 5, "nb", 1, 4,
log_marg = FALSE,iter=iterations)
boxplot_post_dist(stan_fit, "k", ages, years)
boxplot_post_dist(stan_fit, "g", ages, years)

---

Bayesian Cairns-Blake-Dowd (CBD) model with Stan

Description

Fit and Forecast Bayesian CBD model. The model can be fitted with a Poisson or Negative-Binomial distribution. The function outputs posteriors distributions for each parameter, predicted death rates and log-likelihoods.

Usage

cbd_stan(
  death,
  exposure,
  age,
  forecast,
  validation = 0,
  family = c("poisson", "nb"),
  ...
)

Arguments

death	Matrix of deaths.
exposure	Matrix of exposures.
age	Vector of ages.
forecast	Number of years to forecast.
validation	Number of years for validation.
family	specifies the random component of the mortality model. "Poisson" assumes a Poisson model with log link and "nb" assumes a negative-binomial model with log link and overdispersion parameter $\phi$.
...	Arguments passed to rstan::sampling (e.g. iter, chains).

Details

The created model is either a log-Poisson or a log-Negative-Binomial version of the CBD model:

$D_{x,t} \sim \mathcal{P}(\mu_{x,t} e_{x,t})$

or

$D_{x,t}\sim NB\left(\mu_{x,t} e_{x,t},\phi\right)$

with

$\log \mu_{xt} = \kappa_t^{(1)} + (x-\bar{x})\kappa_t^{(2)},$

where $\bar{x}$ is the average age in the data.

For the period terms, we consider a multivariate random walk with drift:

$\boldsymbol{\kappa}_{t}=\boldsymbol{c}+\boldsymbol{\kappa}_{t-1}+\boldsymbol{\epsilon}_{t}^{\kappa},\quad \bm{\kappa}_{t}=\left(\begin{array}{c}\kappa_{t}^{(1)} \\\kappa_{t}^{(2)}\end{array}\right), \quad \boldsymbol{\epsilon}_{t}^{\kappa} \sim N\left(\mathbf{0}, \Sigma\right),$

with normal priors: $\boldsymbol{c} \sim N(0,10)$.

The variance-covariance matrix of the error term is defined by

$\boldsymbol{\Sigma}=\left(\begin{array}{cc}\sigma_1^{2} & \rho_{\Sigma} \sigma_1 \sigma_2 \\\rho_{\Sigma} \sigma_1 \sigma_{Y} & \sigma_2^{2}\end{array}\right)$

where the variance coefficients have independent exponential priors: $\sigma_1, \sigma_2 \sim Exp(0.1)$ and the correlation parameter has a uniform prior: $\rho_{\Sigma} \sim U\left[-1,1\right]$. As for the other models, the overdispersion parameter has a prior distribution given by

$\frac{1}{\phi} \sim Half-N(0,1).$

Value

An object of class stanfit returned by rstan::sampling

References

Cairns, A. J. G., Blake, D., & 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

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-NB model
ages.fit<-50:90
years.fit<-1970:2017
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-50 # Toy example, consider at least 2000 iterations
fitCBD=cbd_stan(death = deathFR,exposure=exposureFR, age=ages.fit, forecast = 10,
family = "poisson",iter=iterations,chains=1)

---

Compute the model evidence/marginal likelihood via bridge sampling (or via the harmonic mean estimator if bridge sampling fails)

Description

Compute the model evidence/marginal likelihood via bridge sampling (or via the harmonic mean estimator if bridge sampling fails)

Usage

compute_weights_BMA(stan_fits, mortality_models)

Arguments

stan_fits	list of Stan model fits where the marginal likelihood was computed via bridge sampling
mortality_models	vector of mortality models names

Value

data frame with model evidence for BMA

---

Function to get the a posterior means of the parameters based on a stanfit object

Description

Function to get the a posterior means of the parameters based on a stanfit object

Usage

extract_map(stan_fit)

Arguments

stan_fit

a stanfit object

Value

named list with the point estimates of the parameters

---

Wrapper function to fit and forecast mortality models

Description

Wrapper function to fit and forecast mortality models

Usage

fit_mo_mo(
  mortality_model = "lc",
  death,
  exposure,
  ages = 50:90,
  validation = 0,
  forecast = 1,
  family = "nb",
  chains = 1,
  cores = 4,
  log_marg = FALSE,
  iter = 2000
)

Arguments

mortality_model	name of the mortality model
death	death matrix
exposure	exposure matrix
ages	vector of ages
validation	size of the validation set
forecast	number of calendar years to be forecast
family	underlying count distribution
chains	number of Markov chains
cores	number of cores used
log_marg	Do we compute the marginal likelihood or not?
iter	Length of the Markov chain trajectory

Value

a stanfit object

Examples

years <- 1990:2017
ages <- 50:90
cohorts <- sort(unique(as.vector(sapply(years, function(year) year - ages))))
death <- FRMaleData$Dxt[formatC(ages),formatC(years)]
exposure <- FRMaleData$Ext[formatC(ages),formatC(years)]
stan_fit <- fit_mo_mo("m6", death , exposure, ages, 0, 5, "nb", 1, 4,
log_marg = FALSE,iter=50)
boxplot_post_dist(stan_fit, "k", ages, years)
boxplot_post_dist(stan_fit, "g", ages, years)

---

Deaths and Exposures Data of French Males

Description

List containing deaths and exposures of French Male for ages 0-110 and years 1816-2017.

Usage

FRMaleData

Format

A list of 2 matrices

Dxt

Matrix of deaths, with 111 rows (ages) and 202 columns (years)

Ext

Matrix of exposures, with 111 rows (ages) and 202 columns (years)

Source

Human Mortality Database https://www.mortality.org/

References

Human Mortality Database (2011). University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at https://www.mortality.org/

---

Bayesian Lee-Carter with Stan

Description

Fit and Forecast Bayesian Lee-Carter model. The model can be fitted with a Poisson or Negative-Binomial distribution. The function outputs posteriors distributions for each parameter, predicted death rates and log-likelihoods.

Usage

lc_stan(
  death,
  exposure,
  forecast,
  validation = 0,
  family = c("poisson", "nb"),
  ...
)

Arguments

death	Matrix of deaths.
exposure	Matrix of exposures.
forecast	Number of years to forecast.
validation	Number of years for validation.
family	specifies the random component of the mortality model. "Poisson" assumes a Poisson model with log link and "nb" assumes a negative-binomial model with log link and overdispersion parameter $\phi$.
...	Arguments passed to rstan::sampling (e.g. iter, chains).

Details

The created model is either a log-Poisson or a log-Negative-Binomial version of the Lee-Carter model:

$D_{x,t} \sim \mathcal{P}(\mu_{x,t} e_{x,t})$

or

$D_{x,t}\sim NB\left(\mu_{x,t} e_{x,t},\phi\right)$

with

$\log \mu_{xt} = \alpha_x + \beta_x\kappa_t.$

To ensure the identifiability of th model, we impose

$\sum_x\beta_x = 1,\kappa_1=0.$

For the priors, the model chooses relatively wide priors:

$\alpha_x \sim N(0,100),\beta_{x} \sim Dir(1,\dots,1),\frac{1}{\phi} \sim Half-N(0,1).$

For the period term, we consider a first order autoregressive process (AR(1)) with linear trend:

$\kappa_{t}=c+\kappa_{t-1}+\epsilon_{t},\epsilon_{t}\sim N(0,\sigma^2)$

with $c \sim N(0,10),\sigma \sim Exp(0.1)$.

Value

An object of class stanfit returned by rstan::sampling.

References

Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(419), 659-671.

Examples

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-Poisson model
ages.fit<-50:90
years.fit<-1970:2017
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-50 # Toy example, consider at least 2000 iterations
fitLC=lc_stan(death = deathFR,exposure=exposureFR, forecast = 10,
family = "poisson",iter=iterations,chains=1)

---

Bayesian M6 model with Stan

Description

Fit and Forecast Bayesian M6 model (CBD with cohort effect) introduced in Cairns et al (2009). The model can be fitted with a Poisson or Negative-Binomial distribution. The function outputs posteriors distributions for each parameter, predicted death rates and log-likelihoods.

Usage

m6_stan(
  death,
  exposure,
  forecast,
  age,
  validation = 0,
  family = c("poisson", "nb"),
  ...
)

Arguments

death	Matrix of deaths.
exposure	Matrix of exposures.
forecast	Number of years to forecast.
age	Vector of ages.
validation	Number of years for validation.
family	specifies the random component of the mortality model. "Poisson" assumes a Poisson model with log link and "nb" assumes a negative-binomial model with log link and overdispersion parameter $\phi$.
...	Arguments passed to rstan::sampling (e.g. iter, chains).

Details

The created model is either a log-Poisson or a log-Negative-Binomial version of the M6 model:

$D_{x,t} \sim \mathcal{P}(\mu_{x,t} e_{x,t})$

or

$D_{x,t}\sim NB\left(\mu_{x,t} e_{x,t},\phi\right)$

with

$\log \mu_{xt} = \kappa_t^{(1)} + (x-\bar{x})\kappa_t^{(2)}+\gamma_{t-x},$

where $\bar{x}$ is the average age in the data.

To ensure the identifiability of th model, we impose

$\gamma_1=0,\gamma_C=0,$

where $C$ represents the most recent cohort in the data.

For the period terms, we consider a multivariate random walk with drift:

$\boldsymbol{\kappa}_{t}=\boldsymbol{c}+ \boldsymbol{\kappa}_{t-1}+\boldsymbol{\epsilon}_{t}^{\kappa},\quad \boldsymbol{\kappa}_{t}=\left(\begin{array}{c}\kappa_{t}^{(1)} \\\kappa_{t}^{(2)}\end{array}\right), \quad \boldsymbol{\epsilon}_{t}^{\kappa} \sim N\left(\mathbf{0}, \Sigma\right),$

with normal priors: $\boldsymbol{c} \sim N(0,10)$.

The variance-covariance matrix of the error term is defined by

$\boldsymbol{\Sigma}=\left(\begin{array}{cc}\sigma_1^{2} & \rho_{\Sigma} \sigma_1 \sigma_2 \\\rho_{\Sigma} \sigma_1 \sigma_{Y} & \sigma_2^{2}\end{array}\right)$

where the variance coefficients have independent exponential priors: $\sigma_1, \sigma_2 \sim Exp(0.1)$ and the correlation parameter has a uniform prior: $\rho_{\Sigma} \sim U\left[-1,1\right]$. As for the other models, the overdispersion parameter has a prior distribution given by

$\frac{1}{\phi} \sim Half-N(0,1).$

For the cohort term, we consider a second order autoregressive process (AR(2)):

$\gamma_{c}=\psi_1 \gamma_{c-1}+\psi_2 \gamma_{c-2}+\epsilon^{\gamma}_{t},\quad \epsilon^{\gamma}_{t}\sim N(0,\sigma_{\gamma}).$

To close the model specification, we impose some vague priors assumptions on the hyperparameters:

$\psi_1,\psi_2 \sim N(0,10),\quad \sigma_{\gamma}\sim Exp(0.1).$

Value

An object of class stanfit returned by rstan::sampling.

References

Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., Ong, A., & 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

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-Poisson model
ages.fit<-70:90
years.fit<-1990:2010
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-50 # Toy example, consider at least 2000 iterations
fitM6=m6_stan(death = deathFR,exposure=exposureFR, age=ages.fit,forecast = 5,
family = "poisson",iter=iterations,chains=1)

---

Model averaging/weighting via future-out stacking or pseudo-BMA weighting

Description

Mortality Model averaging via stacking of predictive distributions and Pseudo-BMA weighting. Based on Yao et al. (2018) but adapted by Barigou et al. (2021) for mortality forecasting.

Usage

mortality_weights(X)

Arguments

X	A list of stanfit objects.

Details

Mortality model averaging via stacking of predictive distributions or pseudo-BMA weighting. Both approaches were proposed in Yao et al. (2018) based leave-one-out cross-validation which is not suited for forecasting. Barigou et al. (2021) adapted both approaches based on leave-future-out validation which is more appropriate for mortality forecasting.

The stacking method combines all models by maximizing the leave-future-out predictive density of the combination distribution. That is, it finds the optimal linear combining weights for maximizing the leave-future-out log score.

The pseudo-BMA method finds the relative weights proportional to the expected log predictive density of each model.

Similar to Yao et al. (2018), we recommend stacking for averaging predictive distributions as pseudo-BMA tends to select only one model.

Value

A matrix containing one weight for each model and each approach.

References

Yao, Y., Vehtari, A., Simpson, D., & Gelman, A. (2018). Using stacking to average Bayesian predictive distributions (with discussion). Bayesian Analysis, 13(3), 917-1007.

Barigou K., Goffard P-O., Loisel S., Salhi Y. (2021). Bayesian model averaging for mortality forecasting using leave-future-out validation. arXiv preprint arXiv:2103.15434.

Examples

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-Poisson model
#where the 10 last years are held out for validation. We search for the model weights between
#the Lee-Carter model and the RH model (Lee-Carter with cohort effect).
ages.fit<-50:90
years.fit<-1970:2017
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-1000 # Toy example, consider at least 2000 iterations
fitLC=lc_stan(death = deathFR,exposure=exposureFR, forecast = 10, validation=10,
family = "poisson",iter=iterations,chains=1)
fitRH=rh_stan(death = deathFR,exposure=exposureFR, forecast = 10, validation=10,
family = "poisson",iter=iterations,chains=1)
model_weights<-mortality_weights(list(fitLC,fitRH))

---

Bayesian Renshaw-Haberman model with Stan

Description

Fit and Forecast Bayesian Renshaw-Haberman model (Lee-Carter with cohort effect) introduced in Renshaw and Haberman (2006). The model can be fitted with a Poisson or Negative-Binomial distribution. The function outputs posteriors distributions for each parameter, predicted death rates and log-likelihoods.

Usage

rh_stan(
  death,
  exposure,
  forecast,
  validation = 0,
  family = c("poisson", "nb"),
  ...
)

Arguments

death	Matrix of deaths.
exposure	Matrix of exposures.
forecast	Number of years to forecast.
validation	Number of years for validation.
family	specifies the random component of the mortality model. "Poisson" assumes a Poisson model with log link and "nb" assumes a negative-binomial model with log link and overdispersion parameter $\phi$.
...	Arguments passed to rstan::sampling (e.g. iter, chains).

Details

The created model is either a log-Poisson or a log-Negative-Binomial version of the Renshaw-Haberman model:

$D_{x,t} \sim \mathcal{P}(\mu_{x,t} e_{x,t})$

or

$D_{x,t}\sim NB\left(\mu_{x,t} e_{x,t},\phi\right)$

with

$\log \mu_{xt} = \alpha_x + \beta_x\kappa_t+\gamma_{t-x}.$

To ensure the identifiability of th model, we impose

$\kappa_1=0, \gamma_1=0,\sum gamma_i =0, \gamma_C=0,$

where $C$ represents the most recent cohort in the data.

For the priors, the model chooses wide priors:

$\alpha_x \sim N(0,100),\beta_{x} \sim Dir(1,\dots,1),\frac{1}{\phi} \sim Half-N(0,1).$

For the period term, we consider the standard random walk with drift:

$\kappa_{t}=c+\kappa_{t-1}+\epsilon_{t},\epsilon_{t}\sim N(0,\sigma^2)$

with $c \sim N(0,10),\sigma \sim Exp(0.1)$.

For the cohort term, we consider a second order autoregressive process (AR(2)):

$\gamma_{c}=\psi_1 \gamma_{c-1}+\psi_2 \gamma_{c-2}+\epsilon^{\gamma}_{t},\quad \epsilon^{\gamma}_{t}\sim N(0,\sigma_{\gamma}).$

To close the model specification, we impose some vague priors assumptions on the hyperparameters:

$\psi_1,\psi_2 \sim N(0,10),\quad \sigma_{\gamma}\sim Exp(0.1).$

Value

An object of class stanfit returned by rstan::sampling.

References

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.

Examples

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-Poisson model
ages.fit<-70:90
years.fit<-1990:2010
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-50 # Toy example, consider at least 2000 iterations
fitRH=rh_stan(death = deathFR,exposure=exposureFR, forecast = 5, family = "poisson",
iter=iterations,chains=1)

---

Simulation of death counts from the Age-Period-Cohort mortality model

Description

Simulation of death counts from the Age-Period-Cohort mortality model

Usage

sim_death_apc(a, k, g, phi, years, ages, exposure)

Arguments

a	vector of age component
k	vector of period component
g	vector of cohort component
phi	dispersion parameter
years	vector of calendar years
ages	vectors of ages
exposure	matrix of exposure data

Value

matrix of death count

---

Simulation of death counts from the CBD model

Description

Simulation of death counts from the CBD model

Usage

sim_death_cbd(k, k2, phi, years, ages, exposure)

Arguments

k	first vector of period component
k2	second vector of period component
phi	dispersion parameter
years	vector of calendar years
ages	vectors of ages
exposure	matrix of exposure data

Value

matrix of death count

---

Simulation of death counts from the Lee-Carter mortality model

Description

Simulation of death counts from the Lee-Carter mortality model

Usage

sim_death_lc(a, b, k, phi, exposure)

Arguments

a	vector of age component
b	vector of age/year component
k	vector of year component
phi	dispersion parameter
exposure	matrix of exposure data

Value

matrix of death count

---

Simulation of death counts from the M6 model

Description

Simulation of death counts from the M6 model

Usage

sim_death_m6(k, k2, g, phi, years, ages, exposure)

Arguments

k	first vector of period component
k2	second vector of period component
g	vector of cohort component
phi	dispersion parameter
years	vector of calendar years
ages	vectors of ages
exposure	matrix of exposure data

Value

matrix of death count

---

Simulation of death counts from a hybrid model that averages the mortality rates from the cbd and rh models

Description

Simulation of death counts from a hybrid model that averages the mortality rates from the cbd and rh models

Usage

sim_death_mix_cbd_rh(params_cbd, params_rh, years, ages, exposure, q)

Arguments

params_cbd	named list that contains the parameters of the cbd model
params_rh	named list that contains the parameters of the rh model
years	vector of calendar year
ages	vector of ages
exposure	matrix of exposure data
q	mixing parameter (0 <- rh, 1 <- cbd)

Value

matrix of death count

---

Simulation of death counts from the Renshaw-Haberman mortality model

Description

Simulation of death counts from the Renshaw-Haberman mortality model

Usage

sim_death_rh(a, b, k, g, phi, years, ages, exposure)

Arguments

a	vector of age component
b	vector of age/year component
k	vector of period component
g	vector of cohort component
phi	dispersion parameter
years	vector of calendar years
ages	vectors of ages
exposure	matrix of exposure data

Value

matrix of death count

---

Simulation of mortality data from various models

Description

Simulation of mortality data from various models

Usage

sim_mortality_data(a, k, k2, b, g, phi, years, ages, exposure, mortality_model)

Arguments

a	vector of age component
k	first vector of time component
k2	second vector of time component
b	vector of age/time component
g	vector of cohort component
phi	dispersion parameter
years	vector of calendar year
ages	vector of ages
exposure	matrix of exposure
mortality_model	name of the mortality model that we simulate from

Value

matrix of death counts

---