cbd_stan {StanMoMo}

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)