blc {BayesMortalityPlus} R Documentation

## Lee-Carter Bayesian Estimation for mortality data

### Description

Performs Bayesian estimation of the Lee-Carter model considering different variances for each age.

### Usage

blc(Y, prior = NULL, init = NULL, M = 2000, bn = 1000)


### Arguments

 Y Log-mortality rates for each age. prior A list containing the prior mean m_0 and the prior variance C_0. init A list containing initial values for \alpha, \beta, \phi_V, \phi_W and \theta. M The number of iterations. bn The number of initial iterations from the Gibbs sampler that should be discarded (burn-in). Default considers half the number of iterations.

### Details

Let Y_{it} be the log mortality rate at age i and time t. The Lee-Carter model is specified as follows:

Y_{it} = \alpha_i + \beta_i \kappa_t + \varepsilon_{it}, i=1,...,p and t=1,...,T,

where \alpha = (\alpha_1,...,\alpha_p)' are the interept of the model that represent the log-mortality rate mean in each age; \beta = (\beta_1,...,\beta_p)' are the coefficient regression that represent the speed of relative change in the log-mortality rate in each age. \kappa = (\kappa_1,...,\kappa_T)' are the state variable that represent the global relative change in log-mortality rate. Finally, \varepsilon_{it} ~ N(0, \sigma^2_i) is the random error.

For the state variable \kappa_t Lee and Carter (1992) proposed a random walk with drift to govern the dynamics over time:

\kappa_t = \kappa_{t-1} + \theta + \omega_t,

where \theta is the drift parameter and \omega_t is the random error of the random walk.

We implement the Bayesian Lee Carter (BLC) model, proposed by Pedroza (2006), to estimate the model. In this approach, we take advantage of the fact that the Bayesian Lee Carter can be specified as dynamic linear model, to estimate the state variables \kappa_t through FFBS algorithm. To estimate the others parameters we use Gibbs sampler to sample from their respective posterior distribution.

### Value

A BLC object.

 alpha Posterior sample from alpha parameter. beta Posterior sample from beta parameter. phiv Posterior sample from phiv parameter. phiv is the precision of the random error of the Lee Carter model. theta Posterior sample from theta. phiw Posterior sample from phiw. phiw is the precision of the random error of the random walk. kappa Sampling from the state variables. Y Y Log-mortality rates for each age passed by the user to fit the model. bn The warmup of the algorithm specified by the user to fit the model. M The number of iterations specified by the user to fit the model. m0 The prior mean of kappa0. C0 The prior covariance matrix of kappa0.

### References

Lee, R. D., & Carter, L. R. (1992). “Modeling and forecasting US mortality.” Journal of the American statistical association, 87(419), 659-671.

Pedroza, C. (2006). “A Bayesian forecasting model: predicting US male mortality.” Biostatistics, 7(4), 530-550.

fitted.BLC(), plot.BLC(), print.BLC() and predict.BLC() for BLC methods to native R functions fitted(), plot(), print() and predict().

expectancy.BLC() and Heatmap.BLC() for BLC methods to compute and visualise the truncated life expectancy via expectancy() and Heatmap() functions.

improvement() to compute the improvement of each age, based on the resulting chains of the beta parameter.

### Examples

## Example of transforming the dataset to fit the function:

## Importing mortality data from the USA available on the Human Mortality Database (HMD):
data(USA)

## Calculating the mortality rates for the general population:
require(dplyr)
require(tidyr)
require(magrittr)

USA %>% mutate(mx = USA$Dx.Total/USA$Ex.Total) -> data

data %>%
filter(Age %in% 18:80 & Year %in% 2000:2015)  %>%
mutate(logmx = log(mx)) %>%
dplyr::select(Year,Age,logmx) %>%
pivot_wider(names_from = Year, values_from = logmx) %>%
dplyr::select(-Age) %>%
as.matrix()  %>%
magrittr::set_rownames(18:80) -> Y

## Fitting the model
fit = blc(Y = Y, M = 100, bn = 20)
print(fit)

## Viewing the results
plot(fit, ages = 18:80)
plot(fit, parameter = "beta", ages=18:80)
improvement(fit)



[Package BayesMortalityPlus version 0.1.1 Index]