Create a new Stochastic Mortality Model

Description

Initialises a StMoMo object which represents a Generalised Age-Period-Cohort Stochastic Mortality Model.

StMoMo.

Usage

StMoMo(link = c("log", "logit"), staticAgeFun = TRUE, periodAgeFun = "NP",
  cohortAgeFun = NULL, constFun = function(ax, bx, kt, b0x, gc, wxt, ages)
  list(ax = ax, bx = bx, kt = kt, b0x = b0x, gc = gc))

Arguments

Details

R implementation of the family of Generalised Age-Period-Cohort stochastic mortality models. This family of models encompasses many models proposed in the literature including the well-known Lee-Carter model, CBD model and APC model.

StMoMo defines an abstract representation of a Generalised Age-Period-Cohort (GAPC) Stochastic model that fits within the general class of generalised non-linear models defined as follows

$D_{xt} \sim Poisson(E_{xt}\mu_{xt}), D_{xt} \sim Binomial(E_{xt},q_{xt})$

$\eta_{xt} = \log \mu_{xt}, \eta_{xt} = \mathrm{logit}\, q_{xt}$

$\eta_{xt} = \alpha_x + \sum_{i=1}^N \beta_x^{(i)}\kappa_t^{(i)} + \beta_x^{(0)}\gamma_{t-x}$

$v: \{\alpha_{x}, \beta_x^{(1)},..., \beta_x^{(N)}, \kappa_t^{(1)},..., \kappa_t^{(N)}, \beta_x^{(0)}, \gamma_{t-x}\} \mapsto \{\alpha_{x}, \beta_x^{(1)},..., \beta_x^{(N)}, \kappa_t^{(1)},..., \kappa_t^{(N)}, \beta_x^{(0)}, \gamma_{t-x}\},$

where

• $\alpha_x$ is a static age function;

• $\beta_x^{(i)}\kappa_t^{(i)}, i = 1,..N$, are age/period terms;

• $\beta_x^{(0)}\gamma_{t-x}$ is the age/cohort term; and

• $v$ is a function defining the identifiability constraints of the model.

Most Stochastic mortality models proposed in the literature can be cast to this representation (See Hunt and Blake (2015)).

Parametric age functions should be scalar functions of the form f <- function(x, ages) taking a scalar age x and a vector of model fitting ages (see examples below).

Do to limitation of functions gnm within package gnm, which is used for fitting "StMoMo" objects to data (see fit.StMoMo), models combining parametric and non-parametric age-modulating functions are not supported at the moment.

Value

A list with class "StMoMo" with components:

References

Plat, R. (2009). On stochastic mortality modeling. Insurance: Mathematics and Economics, 45(3), 393-404.

Hunt, A., & Blake, D. (2015). On the Structure and Classification of Mortality Models Mortality Models. Pension Institute Working Paper. http://www.pensions-institute.org/workingpapers/wp1506.pdf.

See Also

fit.StMoMo, lc, cbd, apc, rh, m6, m7, m8

Examples

#Lee-Carter model
constLC <- function(ax, bx, kt, b0x, gc, wxt, ages) {
 c1 <- mean(kt[1, ], na.rm = TRUE)
 c2 <- sum(bx[, 1], na.rm = TRUE)
 list(ax = ax + c1 * bx, bx = bx / c2, kt = c2 * (kt - c1))
}
LC <- StMoMo(link = "log", staticAgeFun = TRUE, periodAgeFun = "NP",
             constFun = constLC)

plot(fit(LC, data = EWMaleData, ages.fit = 55:89))


#CBD model   
f2 <- function(x, ages) x - mean(ages)
CBD <- StMoMo(link = "logit", staticAgeFun = FALSE,
              periodAgeFun = c("1", f2))
plot(fit(CBD, data = EWMaleData, ages.fit = 55:89))

#Reduced Plat model (Plat, 2009)
f2 <- function(x, ages) mean(ages) - x
constPlat <- function(ax, bx, kt, b0x, gc, wxt, ages) {
 nYears <- dim(wxt)[2]
 x <- ages
 t <- 1:nYears
 c <- (1 - tail(ages, 1)):(nYears - ages[1])
 xbar <- mean(x)
 #nsum g(c)=0, nsum cg(c)=0, nsum c^2g(c)=0
 phiReg <- lm(gc ~ 1 + c + I(c^2), na.action = na.omit)
 phi <- coef(phiReg)
 gc <- gc - phi[1] - phi[2] * c - phi[3] * c^2
 kt[2, ] <- kt[2, ] + 2 * phi[3] * t
 kt[1, ] <- kt[1, ] + phi[2] * t + phi[3] * (t^2 - 2 * xbar * t)
 ax <- ax + phi[1] - phi[2] * x + phi[3] * x^2
 #nsum kt[i, ] = 0
 ci <- rowMeans(kt, na.rm = TRUE)
 ax <- ax + ci[1] + ci[2] * (xbar - x)
 kt[1, ] <- kt[1, ] - ci[1]
 kt[2, ] <- kt[2, ] - ci[2]
 list(ax = ax, bx = bx, kt = kt, b0x = b0x, gc = gc)
}
PLAT <- StMoMo(link = "log", staticAgeFun = TRUE,
               periodAgeFun = c("1", f2), cohortAgeFun = "1",
               constFun = constPlat)

plot(fit(PLAT, data = EWMaleData, ages.fit = 55:89))

#Models not supported
## Not run: 
MnotSup1 <- StMoMo(periodAgeFun = c(f2, "NP"))
MnotSup1 <- StMoMo(periodAgeFun = f2, cohortAgeFun = "NP")

## End(Not run)

Create an Age-Period-Cohort mortality model

Description

Utility function to initialise a StMoMo object representing an Age-Period-Cohort mortality model.

Usage

apc(link = c("log", "logit"))

Arguments

Details

The created model is either a log-Poisson or a logit-Binomial version of the classical age-period-cohort mortality model which has predictor structure

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

To ensure identifiability we follow Cairns et al. (2009) and impose constraints

$\sum_c \gamma_c = 0$

and

$\sum_c c\gamma_c = 0$

Value

An object of class "StMoMo".

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.

See Also

StMoMo, rh

Examples

APC <- apc()
wxt <- genWeightMat(EWMaleData$ages,  EWMaleData$years, clip = 3)
APCfit <- fit(APC, data = EWMaleData, wxt = wxt)
plot(APCfit, parametricbx = FALSE, nCol = 3)

Copy of unexported function arima.string from forecast

Description

Copy of unexported function arima.string from forecast

Usage

arima.string(object, padding = FALSE)

Map Binomial deviance residuals into deaths

Description

This function uses the procedure described in Debon et al. (2010, section 3)

Usage

binomRes2q(dhat, e, res)

Arguments

Value

matching observed deaths

References

Debon, A., Martinez-Ruiz, F., & Montes, F. (2010). A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities. Insurance: Mathematics and Economics, 47(3), 327-336.

Generic method for bootstrapping a fitted Stochastic Mortality Model

Description

bootstrap is a generic function for bootstrapping Stochastic Mortality Models. The function invokes particular methods which depend on the class of the first argument.

Usage

bootstrap(object, nBoot, ...)

Arguments

Details

bootstrap is a generic function which means that new fitting strategies can be added for particular stochastic mortality models. See for instance bootstrap.fitStMoMo.

Bootstrap a fitted Stochastic Mortality Model

Description

Produce bootstrap parameters of a Stochastic Mortality Model to account for parameter uncertainty.

Usage

## S3 method for class 'fitStMoMo'
bootstrap(object, nBoot = 1, type = c("semiparametric",
  "residual"), deathType = c("observed", "fitted"), ...)

Arguments

Details

When type is "residual" the residual bootstrapping approach described in Renshaw and Haberman (2008) is applied, which is an adaptation of the approach of Koissi et al (2006). In the case of a "logit" link with Binomial responses the adaptation described in Debon et al, (2010, section 3) is used.

When type is "semiparametric" the semiparametric approach described in Brouhns et al.(2005) is used. In the case of a "logit" link with Binomial responses a suitable adaptation is applied. If deathType is "observed" then the observed deaths are used in the sampling as in Brouhns et al. (2005) while if deathType is "fitted" the fitted deaths are used in the sampling as in Renshaw and Haberman (2008).

Value

A list with class "bootStMoMo" with components:

References

Brouhns, N., Denuit M., & Van Keilegom, I. (2005). Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Journal, 2005(3), 212-224.

Debon, A., Martinez-Ruiz, F., & Montes, F. (2010). A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities. Insurance: Mathematics and Economics, 47(3), 327-336.

Renshaw, A. E., & Haberman, S. (2008). On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling. Insurance: Mathematics and Economics, 42(2), 797-816.

See Also

simulate.bootStMoMo, plot.bootStMoMo

Examples

#Long computing times
## Not run: 
LCfit <- fit(lc(), data = EWMaleData)

LCResBoot <- bootstrap(LCfit, nBoot = 500, type = "residual")
plot(LCResBoot)
LCSemiObsBoot <- bootstrap(LCfit, nBoot = 500, type = "semiparametric")
plot(LCSemiObsBoot)
LCSemiFitBoot <- bootstrap(LCfit, nBoot = 500, type = "semiparametric",  
                           deathType = "fitted")
plot(LCSemiFitBoot)

## End(Not run)

Create a Cairns-Blake-Dowd mortality model

Description

Utility function to initialise a StMoMo object representing a Cairns-Blake-Dowd mortality model.

Usage

cbd(link = c("logit", "log"))

Arguments

Details

The created model is either a logit-Binomial or a log-Poisson version of the Cairns-Blake-Dowd mortality model which has predictor structure

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

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

Value

An object of class "StMoMo".

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.

See Also

StMoMo, central2initial, m6, m7, m8

Examples

CBD <- cbd()
CBDfit <- fit(CBD, data = central2initial(EWMaleData), ages.fit = 55:89)
plot(CBDfit, parametricbx = FALSE)

Transform StMoMoData from central to initial exposures

Description

Transform StMoMoData from central to initial exposures. Initial exposures are computed by adding one half of the deaths to the central exposures.

Usage

central2initial(data)

Arguments

Value

A StMoMoData object of type "initial".

See Also

initial2central

Examples

CBD <- cbd()
CBDfit <- fit(CBD, data = central2initial(EWMaleData), ages.fit = 55:89)
plot(CBDfit, parametricbx = FALSE)

Extract coefficients from a fitted Stochastic Mortality Model

Description

Extract coefficients from a fitted Stochastic Mortality Model

Usage

## S3 method for class 'fitStMoMo'
coef(object, ...)

Arguments

Value

A list of model parameters with components:

Examples

APCfit <- fit(apc(), data = EWMaleData)
coef(APCfit) 

Binomial deviance

Description

Binomial deviance

Usage

computeDevianceBinomial(obs, fit, exposure, weight)

Arguments

Compute Poisson deviance

Description

Compute Poisson deviance

Usage

computeDeviancePoisson(obs, fit, weight)

Arguments

Compute Binomial loglikelihod

Description

Compute Binomial loglikelihod

Usage

computeLogLikBinomial(obs, fit, exposure, weight)

Arguments

Compute Poisson loglikelihod

Description

Compute Poisson loglikelihod

Usage

computeLogLikPoisson(obs, fit, weight)

Arguments

England and Wales male mortality data

Description

Age-specific deaths and exposures for England and Wales from the Human Mortality Database. This is an object of class StMoMoData.

Usage

EWMaleData

Format

A list with the following components:

Dxt

matrix of deaths data.

Ext

matrix of exposures data (mid year population estimates).

ages

vector of ages.

years

vector of years.

type

the type of exposure in the data (central).

series

name of the extracted seriesin this case males.

label

label of the data.

Details

EWMaleData contains deaths and exposures for England and Wales males for the period 1961-2011 and for ages 0-100. Data taken from the Human Mortality Database on 5 November 2014.

Source

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

References

Human Mortality Database (2014). University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org.

See Also

StMoMoData

Extract the model coefficient

Description

Extract the model coefficient of a stochastic mortality model from a gnm fit of the model

Usage

extractCoefficientsFromGnm(object, coefGnmModel, ages, years, cohorts,
  zeroWeigthAges, zeroWeigthYears, zeroWeigthCohorts)

Arguments

Details

Weight vectors wx, wx, wc are used to identify parameters that cannot be estimated because all the data is weighted out.

Value

A list with the model parameters, ax, bx, kt, b0x, gc

Extract cohort from an age-period array

Description

Extract cohorts from an age-period array. This is useful to construct a life table or to perform actuarial/demographic calculations on a cohort basis using the output of several functions in StMoMo.

Usage

extractCohort(A, age = as.numeric(dimnames(A)[[1]][1]),
  period = as.numeric(dimnames(A)[[2]][1]), cohort = period - age)

Arguments

Value

If the the input array is two dimensional the the output is a a vector with the quantity along the cohort. Otherwise if A is an N-dimensional array the output is an (N-1)-dimensional array with the first dimension representing the cohort.

See Also

fitted.fitStMoMo, forecast.fitStMoMo, simulate.fitStMoMo, simulate.bootStMoMo

Examples

LCfit <- fit(lc(), data = EWMaleData, ages.fit = 55:89)
#Plot forecast mortality rates for the 1950 cohort
LCfor <- forecast(LCfit)
plot(55:61, extractCohort(fitted(LCfit, type = "rates"), cohort = 1950), 
     type = "l", log = "y", xlab = "age", ylab = "Mortality rate", 
     main = "Mortality rates for the 1950 cohort", 
     xlim = c(55,89), ylim = c(0.005, 0.12))
lines(62:89, extractCohort(LCfor$rates, cohort = 1950), lty = 2, col = "blue")


#Plot 10 simulated sets of mortality rates for the cohort 
# aged 60 in year 2010 (i.e., the 1950 cohort)
LCsim <- simulate(LCfit, nsim = 10)
mSim <- extractCohort(LCsim$rates, age = 60, period = 2010)
plot(55:61, extractCohort(fitted(LCfit, type = "rates"), cohort = 1950), 
     type = "l", log = "y", xlab = "age", ylab = "Mortality rate", 
     main = "Mortality rates for the 1950 cohort", 
     xlim = c(55,89), ylim = c(0.005, 0.12))
matlines(62:89, mSim, lty = 2)

Generic for fitting a Stochastic Mortality Model

Description

fit is a generic function for fitting Stochastic Mortality Models. The function invokes particular methods which depend on the class of the first argument.

Usage

fit(object, ...)

Arguments

Details

fit is a generic function which means that new fitting strategies can be added for particular stochastic mortality models. See for instance fit.StMoMo.

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

Description

Fit a Renshaw and Haberman (Lee-Carter with cohorts) mortality model using the iterative Newton-Raphson procedure presented in Algorithm 1 of Hunt and Villegas (2015). This approach helps solve the well-known robustness and converges issues of the Lee-Carter model with cohort-effects.

Usage

## S3 method for class 'rh'
fit(object, data = NULL, Dxt = NULL, Ext = NULL,
  ages = NULL, years = NULL, ages.fit = NULL, years.fit = NULL,
  oxt = NULL, wxt = NULL, start.ax = NULL, start.bx = NULL,
  start.kt = NULL, start.b0x = NULL, start.gc = NULL, verbose = TRUE,
  tolerance = 1e-04, iterMax = 10000, ...)

Arguments

Value

References

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

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)

Fit a Stochastic Mortality Model

Description

Fit a Stochastic Mortality Model to a given data set. The fitting is done using package gnm.

Usage

## S3 method for class 'StMoMo'
fit(object, data = NULL, Dxt = NULL, Ext = NULL,
  ages = NULL, years = NULL, ages.fit = NULL, years.fit = NULL,
  oxt = NULL, wxt = NULL, start.ax = NULL, start.bx = NULL,
  start.kt = NULL, start.b0x = NULL, start.gc = NULL, verbose = TRUE,
  ...)

Arguments

Details

Fitting is done using function gnm within package gnm. This is equivalent to minimising (maximising) the deviance (log-likelihood) of the model. Ages and years in the data should be of type numeric. Data points with zero exposure are assigned a zero weight and are ignored in the fitting process. Similarly, NA are assigned a zero weight and ignored in the fitting process. Parameter estimates can be plotted using function plot.fitStMoMo.

Value

A list with class "fitStMoMo" with components:

@seealso StMoMoData, genWeightMat, plot.fitStMoMo, EWMaleData

Examples

   

# Lee-Carter model only for older ages
LCfit <- fit(lc(), data = EWMaleData, ages.fit = 55:89)
plot(LCfit)

# Use arguments Dxt, Ext, ages, years to pass fitting data
LCfit <- fit(lc(), Dxt = EWMaleData$Dxt, Ext = EWMaleData$Ext, 
             ages = EWMaleData$ages, years = EWMaleData$years, 
             ages.fit = 55:89)
plot(LCfit)

# APC model weigthing out the 3 first and last cohorts
wxt <- genWeightMat(EWMaleData$ages,  EWMaleData$years, clip = 3)
APCfit <- fit(apc(), data = EWMaleData, wxt = wxt)
plot(APCfit, parametricbx = FALSE, nCol = 3)

# Set verbose = FALSE for silent fitting
APCfit <- fit(apc(), data = EWMaleData,  wxt = wxt, 
              verbose = FALSE)
## Not run: 
  # Poisson Lee-Carter model with the static age function set to  
  # the mean over time of the log-death rates
  constLCfix_ax <- function(ax, bx, kt, b0x, gc, wxt, ages){  
    c1 <- sum(bx, na.rm = TRUE)
    bx <- bx / c1
    kt <- kt * c1  
    list(ax = ax, bx = bx, kt = kt, b0x = b0x, gc = gc)  
  }  
  LCfix_ax <- StMoMo(link = "log", staticAgeFun = FALSE, 
                     periodAgeFun = "NP", constFun =  constLCfix_ax)
  LCfix_axfit <- fit(LCfix_ax, data= EWMaleData, 
                     oxt = rowMeans(log(EWMaleData$Dxt / EWMaleData$Ext)))
  plot(LCfix_axfit)

## End(Not run)

Compute fitted values for a Stochastic Mortality Model

Description

Returns fitted values for the data used in fitting a Stochastic Mortality Model.

Usage

## S3 method for class 'fitStMoMo'
fitted(object, type = c("link", "rates", "deaths"), ...)

Arguments

Value

A matrix with the fitted values.

Examples

LCfit <- fit(lc(), data = EWMaleData, ages.fit = 55:89)
matplot(LCfit$ages, fitted(LCfit), type = "l", lty = 1, 
        col = rainbow(length(LCfit$years)), xlab = "year", 
        ylab = "log death rate", main = "Fitted rates")

uxthat <- fitted(LCfit, type = "rates")
uxt <- LCfit$Dxt / LCfit$Ext
plot(LCfit$years, uxt["65", ], xlab = "year", ylab = "death rate",
     main = "fitted vs. observed rates at age 65")
lines(LCfit$years, uxthat["65", ])

Forecast mortality rates using a Stochastic Mortality Model

Description

Forecast mortality rates using a Stochastic Mortality Model fit. The period indexes $\kappa_t^{(i)}, i = 1,..N,$ are forecasted using ether a Multivariate Random Walk with Drift (MRWD) or $N$ independent ARIMA$(p, d, q)$ models. The cohort index $\gamma_{t-x}$ is forecasted using an ARIMA$(p, d, q)$. By default an ARIMA$(1, 1, 0)$ with a constant is used.

Usage

## S3 method for class 'fitStMoMo'
forecast(object, h = 50, level = c(80, 95),
  oxt = NULL, gc.order = c(1, 1, 0), gc.include.constant = TRUE,
  jumpchoice = c("fit", "actual"), kt.method = c("mrwd", "iarima"),
  kt.order = NULL, kt.include.constant = TRUE, kt.lookback = NULL,
  gc.lookback = NULL, ...)

Arguments

Details

If kt.method is "mrwd", fitting and forecasting of the time series model for the period indexes is done with a Multivariate Random Walk with Drift using the function mrwd.

If kt.method is "iarima", fitting and forecasting of the time series model for the period indexes is done with $N$ independent arima models using the function iarima. See this latter function for details on input arguments kt.order and kt.include.constant.

Fitting and forecasting of the ARIMA model for the cohort index is done with function Arima from package forecast. See the latter function for further details on input arguments gc.order and gc.include.constant.

Note that in some cases forecast of the cohort effects may be needed for a horizon longer than h. This is the case when in the fitted model the most recent cohorts have been zero weighted. The forecasted cohorts can be seen in gc.f$cohorts.

Value

A list of class "forStMoMo" with components:

Examples

#Lee-Carter (random walk with drift)
LCfit <- fit(lc(), data = EWMaleData, ages.fit = 55:89)
LCfor <- forecast(LCfit)
plot(LCfor)

#Lee-Carter (forecast with ARIMA(1,1,2) with drift )
LCfor.iarima1 <- forecast(LCfit, kt.method = "iarima", kt.order = c(1, 1 , 2))
plot(LCfor.iarima1)

#Lee-Carter (forecast with auto.arima)
LCfor.iarima2 <- forecast(LCfit, kt.method = "iarima")
plot(LCfor.iarima2)

#CBD (Multivariate random walk with drift)
CBDfit <- fit(cbd(), data = central2initial(EWMaleData), ages.fit = 55:89)
CBDfor <- forecast(CBDfit)
plot(CBDfor, parametricbx = FALSE)

#CBD (Independent Arima models)
kt.order <- matrix(c(1, 1, 2,  #ARIMA(1, 1, 2) for k[1]
                     0, 1, 1), #ARIMA(0, 1, 1) for k[2]
                   nrow = 2, ncol = 3, byrow = TRUE)
CBDfor.iarima <- forecast(CBDfit, kt.method = "iarima", kt.order = kt.order)
plot(CBDfor.iarima, parametricbx = FALSE)

#APC: Compare forecast with different models for the cohort index
wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
APCfit <- fit(apc(), data = EWMaleData,  ages.fit = 55:89, 
              wxt = wxt)
APCfor1 <- forecast(APCfit)
plot(APCfor1, parametricbx = FALSE, nCol = 3)
APCfor2 <- forecast(APCfit, gc.order = c(0, 2, 2))
plot(APCfor2, only.gc = TRUE)
plot(c(APCfit$years, APCfor1$years), 
     cbind(APCfor1$fitted, APCfor1$rates)["65", ], 
     type = "l", xlab = "year", ylab = "Mortality rate at age 65", 
     main = "Forecasts with different models for gc")
lines(APCfor2$years, APCfor2$rates["65", ], col = "blue")
points(APCfit$years, (APCfit$Dxt / APCfit$Ext)["65", ], pch = 19) 

#Compare Lee-Carter forecast using: 
#  1. Fitted jump-off rates and all history for kt
#  2. Actual jump-off rates and all history for kt
#  3. Fitted jump-off rates and only history for 
#     the past 30 years of kt (i.e 1982-2011)

LCfor1 <- forecast(LCfit)
LCfor2 <- forecast(LCfit, jumpchoice = "actual")
LCfor3 <- forecast(LCfit, kt.lookback = 30) 

plot(LCfit$years, (LCfit$Dxt / LCfit$Ext)["60", ],
     xlim = range(LCfit$years, LCfor1$years),
     ylim = range((LCfit$Dxt / LCfit$Ext)["60", ], LCfor1$rates["60", ],
                  LCfor2$rates["60", ], LCfor3$rates["60", ]),
     type = "p", xlab = "year", ylab = "rate",
     main = "Lee-Carter: Forecast of mortality rates at age 60")
lines(LCfit$years, fitted(LCfit, type = "rates")["60", ])
lines(LCfor1$years, LCfor1$rates["60", ], lty = 2)
lines(LCfor2$years, LCfor2$rates["60", ], lty = 3, col = "blue")
lines(LCfor3$years, LCfor3$rates["60", ], lty = 4, col = "red")
legend("topright",legend = c("Fitted jump-off", "Actual jump-off", 
       "Fitted jump-off, 30 year look-back"), 
       lty = 1:3, col = c("black", "blue", "red"))

Forecast independent arima series

Description

Returns forecasts and other information for a group of independent arima series.

Usage

## S3 method for class 'iarima'
forecast(object, h = 10, level = c(80, 95), fan = FALSE,
  ...)

Arguments

Value

An object of class "iarimaForecast" with components:

@export

Forecast a Multivariate Random Walk with Drift

Description

Returns forecasts and other information for a Multivariate Random Walk with Drift model.

Usage

## S3 method for class 'mrwd'
forecast(object, h = 10, level = c(80, 95), fan = FALSE,
  ...)

Arguments

Value

An object of class "mrwdForecast" with components:

@export

Generate Binomial residual bootstrap sample

Description

Generate Binomial bootstrap samples using the procedure described in Debon et al. (2010, section 3)

Usage

genBinomialResBootSamples(devRes, dhat, E, nBoot)

Arguments

Value

a list of length nBoot of matrices with matching sampled deaths

References

Debon, A., Martinez-Ruiz, F., & Montes, F. (2010). A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities. Insurance: Mathematics and Economics, 47(3), 327-336.

Generate Binomial semiparametric bootstrap samples

Description

Generate Binomial semiparametric bootstrap samples using a suitable adaptation of the Poisson procedure described in Brouhns et al (2005).

Usage

genBinomialSemiparametricBootSamples(D, E, nBoot)

Arguments

Value

a list of length nBoot of matrices with sampled deaths

References

Brouhns, N., Denuit M., & Van Keilegom, I. (2005). Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Journal, 2005(3), 212-224.

Generate Poisson residual bootstrap sample

Description

Generate Poisson bootstrap samples using the procedure described in Renshaw and Haberman (2008)

Usage

genPoissonResBootSamples(devRes, dhat, nBoot)

Arguments

Value

a list of length nBoot of matrices with matching sampled deaths

References

Renshaw, A. E., & Haberman, S. (2008). On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling. Insurance: Mathematics and Economics, 42(2), 797-816.

Generate Poisson semiparametric bootstrap samples

Description

Generate Poisson semiparametric bootstrap samples using the procedure described in Brouhns et al (2005).

Usage

genPoissonSemiparametricBootSamples(D, nBoot)

Arguments

Value

a list of length nBoot of matrices with sampled deaths

References

Brouhns, N., Denuit M., & Van Keilegom, I. (2005). Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Journal, 2005(3), 212-224.

Generate weight matrix

Description

Generates a weight matrix given a group of ages and years and a set of cohorts which are to be given zero weight. This is useful for excluding some data points when fitting a Stochastic Mortality Model (see fit.StMoMo).

Usage

genWeightMat(ages, years, clip = 0, zeroCohorts = NULL)

Arguments

Value

A 0-1 matrix with 0 for the zero-weighed cohorts.

See Also

fit.StMoMo

Examples

#Zero-weight the first three and last three cohorts
wxt1 <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
APCfit1 <- fit(apc(), data = EWMaleData, ages.fit = 55:89, 
               wxt = wxt1)
plot(APCfit1, parametricbx = FALSE, nCol = 3)

#Also Zero-weight the 1886 cohort
wxt2 <- genWeightMat(55:89,  EWMaleData$years, clip = 3, 
                     zeroCohorts = 1886)
APCfit2 <- fit(apc(), data = EWMaleData, ages.fit = 55:89, 
               wxt = wxt2)
plot(APCfit2, parametricbx = FALSE, nCol = 3)

Extract a lighter version of a fitted Stochastic Mortality Model

Description

Obtain a lighter version of a fitted Stochastic Mortality Model with the essential information for plotting, forecasting, and simulation.

Usage

getMinimalFitStMoMo(object)

Arguments

Value

A list with class "fitStMoMo" with components

Fit independent arima series to a multivariate time series

Description

Fits independent arima series to x, a multivariate time series.

Usage

iarima(x, order = NULL, include.constant = TRUE, ...)

Arguments

Details

The fitting of the ARIMA models for each time series is done with function Arima from package forecast. See the latter function for further details on input arguments kt.order and kt.include.constant.

Value

an object of class "iarima" with components:

Transform StMoMoData from initial to central exposures

Description

Transform StMoMoData from initial to central exposures. Central exposures are computed by substracting one half of the deaths from the initial exposures.

Usage

initial2central(data)

Arguments

Value

A StMoMoData object of type "central".

See Also

central2initial

Inverse Logit function invlogit computes the inverse logit function

Description

Inverse Logit function invlogit computes the inverse logit function

Usage

invlogit(x)

Create a Lee-Carter model

Description

Utility function to initialise a StMoMo object representing a Lee-Carter model.

Usage

lc(link = c("log", "logit"), const = c("sum", "last", "first"))

Arguments

Details

The created model is either a log-Poisson (see Brouhns et al (2002)) or a logit-Binomial version of the Lee-Carter model which has predictor structure

$\eta_{xt} = \alpha_x + \beta_x\kappa_t.$

To ensure identifiability one of the following constraints is imposed

$\sum_t\kappa_t = 0,\,\kappa_1 = 0,\, \kappa_n = 0$

depending on the value of const, and

$\sum_x\beta_x = 1.$

Value

An object of class "StMoMo".

References

Brouhns, N., Denuit, M., & Vermunt, J. K. (2002). A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31(3), 373-393.

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

See Also

StMoMo

Examples

#sum(kt) = 0 and log link
LC1 <- lc()
LCfit1<-fit(LC1, data = EWMaleData, ages.fit = 55:89)
plot(LCfit1)

#kt[1] = 0 and log link
LC2 <- lc(const = "first")
LCfit2<-fit(LC2, data = EWMaleData, ages.fit = 55:89)
plot(LCfit2)

#kt[n] = 0 and logit link
LC3 <- lc("logit", "last")
LCfit3<-fit(LC3, data = EWMaleData, ages.fit = 55:89)
plot(LCfit3)

Logit function logit computes the logit function

Description

Logit function logit computes the logit function

Usage

logit(x)

Log-Likelihood of a fitStMoMo object

Description

Returns the log-likelihood of a fitted Stochastic Mortality Model.

Usage

## S3 method for class 'fitStMoMo'
logLik(object, ...)

Arguments

Value

The log-likelihood of the fitted model.

Create an M6 type extension of the Cairns-Blake-Dowd mortality model

Description

Utility function to initialise a StMoMo object representing the M6 (CBD with cohorts) extension of the Cairns-Blake-Dowd mortality model introduced in Cairns et al (2009).

Usage

m6(link = c("logit", "log"))

Arguments

Details

The created model is either a logit-Binomial or a log-Poisson version of the M6 model which has predictor structure

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

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

Identifiability of the model is accomplished by applying parameters constraints

$\sum_c\gamma_c = 0, \sum_c c\gamma_c = 0$

which ensure that the cohort effect fluctuates around zero and has no linear trend. These constraints are applied using the strategy discussed in Appendix A of Haberman and Renshaw (2011).

Value

An object of class "StMoMo".

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.

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

See Also

StMoMo, central2initial, cbd, m7, m8

Examples

M6 <- m6()
wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
M6fit <- fit(M6, data = central2initial(EWMaleData), ages.fit = 55:89)
plot(M6fit, parametricbx = FALSE)

Create an M7 type extension of the Cairns-Blake-Dowd mortality model

Description

Utility function to initialise a StMoMo object representing the M7 extension of the Cairns-Blake-Dowd mortality model introduced in Cairns et al (2009).

Usage

m7(link = c("logit", "log"))

Arguments

Details

The created model is either a logit-Binomial or a log-Poisson version of the M7 model which has predictor structure

$\eta_{xt} = \kappa_t^{(1)} + (x-\bar{x})\kappa_t^{(2)} + ((x-\bar{x})^2 - \hat{\sigma}^2_x)\kappa_t^{(2)} + \gamma_{t-x},$

where $\bar{x}$ is the average age in the data and $\hat{\sigma}^2_x$ is the average value of $(x-\bar{x})^2$.

Identifiability of the model is accomplished by applying parameters constraints

$\sum_c\gamma_c = 0, \sum_c c\gamma_c = 0, \sum_c c^2\gamma_c = 0$

which ensure that the cohort effect fluctuates around zero and has no linear or quadratic trend. These constraints are applied using the strategy discussed in Appendix A of Haberman and Renshaw (2011).

Value

An object of class "StMoMo".

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.

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

See Also

StMoMo, central2initial, cbd, m6, m8

Examples

M7 <- m7()
wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
M7fit <- fit(M7, data = central2initial(EWMaleData), ages.fit = 55:89)
plot(M7fit, parametricbx = FALSE)

Create an M8 type extension of the Cairns-Blake-Dowd mortality model

Description

Utility function to initialise a StMoMo object representing the M8 extension of the Cairns-Blake-Dowd mortality model introduced in Cairns et al (2009).

Usage

m8(link = c("logit", "log"), xc)

Arguments

Details

The created model is either a logit-Binomial or a log-Poisson version of the M8 model which has predictor structure

$\eta_{xt} = \kappa_t^{(1)} + (x-\bar{x})\kappa_t^{(2)} + (x_c-x)\gamma_{t-x}$

where $\bar{x}$ is the average age in the data and $x_c$ is a predefined constant. Identifiability of the model is accomplished by applying parameters constraint

$\sum_c\gamma_c = 0.$

Value

An object of class "StMoMo".

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.

See Also

StMoMo, central2initial, cbd, m6, m7

Examples

M8 <- m8(xc = 89)
wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
M8fit <- fit(M8, data = central2initial(EWMaleData), ages.fit = 55:89)
plot(M8fit, parametricbx = FALSE)

Fit a Multivariate Random Walk with Drift

Description

Fits a Multivariate Random Walk with Drift to x, a multivariate time series.

Usage

mrwd(x)

Arguments

Details

For further information on the Multivariate Random Walk with drift see Appendix B in Haberman and Renshaw (2011).

Value

an object of class "mrwd" with components:

References

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

Plot bootstrapped parameters of a Stochastic Mortality Model

Description

Plot fancharts of bootstrapped parameters of a Stochastic Mortality Model stored in an object of class "bootStMoMo".

Usage

## S3 method for class 'bootStMoMo'
plot(x, nCol = 2, parametricbx = TRUE, 
                          colour = rgb(0, 0, 0), 
                          probs = c(2.5, 10, 25, 50, 75, 90, 97.5), ...)

Arguments

See Also

plot.fitStMoMo

Examples

#Long computing times
## Not run: 
CBDfit <- fit(cbd(), data = central2initial(EWMaleData),
              ages.fit = 55:89)
CBDResBoot <- bootstrap(CBDfit, nBoot = 500)
plot(CBDResBoot)
plot(CBDResBoot, parametricbx = FALSE, probs = seq(2.5, 97.5, 2.5))

## End(Not run)

Plot fitted parameters from a stochastic mortality model

Description

Plot fitted parameters of a stochastic mortality model of class "fitStMoMo".

Usage

## S3 method for class 'fitStMoMo'
plot(x, nCol = 2, parametricbx = TRUE, type = "l", ...)

Arguments

Examples

#Fit and plot a Lee-Carter model
LCfit <- fit(lc(), data = EWMaleData, ages.fit = 55:89)
plot(LCfit)
plot(LCfit, type = "p", pch = 19)

#Fit and plot a CBD model
CBDfit <- fit(cbd(), data = central2initial(EWMaleData), ages.fit = 55:89)
plot(CBDfit)
plot(CBDfit, parametricbx = FALSE)
plot(CBDfit, nCol = 1, parametricbx = FALSE, lwd = 2)

Plot a forecast from a Stochastic Mortality Model

Description

Plot a forecasted Stochastic Mortality Model of class "forStMoMo".

Usage

## S3 method for class 'forStMoMo'
plot(x, nCol = 2, parametricbx = TRUE, only.kt = FALSE,
 only.gc = FALSE,  colour = "grey60", ...)

Arguments

See Also

plot.fitStMoMo

Examples

wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
APCfit <- fit(apc(), data = EWMaleData, ages.fit = 55:89, wxt = wxt)
APCfor <- forecast(APCfit)
plot(APCfor)
plot(APCfor, parametricbx = FALSE, nCol = 3)
plot(APCfor, only.kt = TRUE)
plot(APCfor, only.gc = TRUE, lwd = 2)

Plot the residuals of a Stochastic Mortality Model

Description

Plots the deviance residuals of a Stochastic Mortality Model which are of class "resStMoMo". Three types of plots are available: scatter plot of residuals by age, period and cohort, colour map (heatmap) of the residuals, and a black and white signplot of the residuals.

Usage

## S3 method for class 'resStMoMo'
plot(x, type = c("scatter", "colourmap", 
                                            "signplot"), 
                                reslim = NULL, plotAge = TRUE, 
                                plotYear = TRUE, plotCohort  = TRUE, 
                                pch = 20, col = NULL, ...)

Arguments

Details

When type = "scatter" scatter plots of the residuals against age, calendar year and cohort (year of birth) are produced.

When type = "colourmap" a two dimensional colour map of the residuals is plotted. This is produced using function image.plot. See image.plot for further parameters that can be passed to this type of plots.

When type = "signplot" a two dimensional black and white map of the residuals is plotted with dark grey representing negative residuals and light grey representing positive residuals. This is produced using function image.default.

@seealso residuals.fitStMoMo

Examples

CBDfit <- fit(cbd(), data = central2initial(EWMaleData), ages.fit = 55:89)
CBDres <- residuals(CBDfit)
plot(CBDres)
plot(CBDres, type = "signplot")
plot(CBDres, type = "colourmap")
  

Plot fanchart of the parameters

Description

Plot fanchart of the parameters

Usage

plotParameterFan(x, y, yBoot, main, xlab, ylab, probs, fan.col, n.fan, ...)

Arguments

Details

In order for the plotting to look appropriately the intervals of the data with non missing values need to be found. Otherwise fanplot::fan doesn't work.

Map poisson deviance residuals into deaths

Description

This function uses the procedure described in Renshaw and Haberman (2008)

Usage

poissonRes2death(dhat, res)

Arguments

Value

matching observed deaths

References

Renshaw, A. E., & Haberman, S. (2008). On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling. Insurance: Mathematics and Economics, 42(2), 797-816.

Predict method for Stochastic Mortality Models fits

Description

Obtain predictions from a Stochastic Mortality Model fit.

Usage

## S3 method for class 'fitStMoMo'
predict(object, years, kt = NULL, gc = NULL,
  oxt = NULL, type = c("link", "rates"), ...)

Arguments

Details

This function evaluates

$\hat{\eta}_{xt} = o_{xt} + \alpha_x + \sum_{i=1}^N \beta_x^{(i)}\kappa_t^{(i)} + \beta_x^{(0)}\gamma_{t-x}$

for a fitted Stochastic Mortality model. In producing a prediction the static age function, $\alpha_x$, and the age-modulating parameters, $\beta_x^{(i)}, i=0, ..., N$, are taken from the fitted model in object while the period indexes, $\kappa_t^{(i)}, i=1,..., N$, and cohort index, $\gamma_{t-x}$, are taken from the function arguments.

This function can be useful, for instance, in producing forecasts of mortality rates using time series models different to those available in forecast.fitStMoMo (See examples below).

Value

A matrix with the predicted values.

See Also

forecast.fitStMoMo

Examples

## Not run: 
##M6 Forecast using VARIMA(1,1) model
library(MTS)

# fit m6
years <- EWMaleData$years
ages.fit <- 55:89
M6 <- m6(link = "log")
M6fit <- fit(M6, data = EWMaleData, ages.fit = ages.fit)

# Forecast kt using VARIMA(1,1) model from MTS
h <- 50
kt.M6 <- t(M6fit$kt) 
kt.M6.diff <- apply(kt.M6, 2, diff)
fit.kt.M6.11 <- VARMA(kt.M6.diff, p = 1, q = 1)
pred.ktdiff.M6.11 <- VARMApred(fit.kt.M6.11, h = h)
pred.kt.M6.11 <- apply(rbind(tail(kt.M6, n = 1),
                             pred.ktdiff.M6.11$pred), 
                       2, cumsum)[-1, ]

# set row names
years.forecast <- seq(tail(years, 1) + 1, length.out = h)
rownames(pred.kt.M6.11) <- years.forecast

# plot kt1
plot(x = c(years, years.forecast),
     y = c(kt.M6[, 1], pred.kt.M6.11[, 1]),
     col = rep(c("black", "red"), times = c(length(years), h)),
     xlab = "time",
     ylab = "k1")

plot(x = c(years, years.forecast),
     y = c(kt.M6[, 2], pred.kt.M6.11[, 2]),
     col = rep(c("black", "red"), times = c(length(years), h)),
     xlab = "time",
     ylab = "k2")

# forecast cohort effect
# the following cohorts are required:
# from 2012 - 89 = 1923
# to 2061 - 55 = 2006
pred.gc.M6 <- forecast(auto.arima(M6fit$gc, max.d = 1), h = h)

# use predict to get rates
pred.qxt.M6.11 <- predict(object = M6fit,
                          years = years.forecast,
                          kt = t(pred.kt.M6.11),
                          gc = c(tail(M6fit$gc, 34), pred.gc.M6$mean),
                          type = "rates")

qxthatM6 <- fitted(M6fit, type = "rates")

# plot mortality profile at age 60, 70 and 80
matplot(1961 : 2061,
        t(cbind(qxthatM6, pred.qxt.M6.11)[c("60", "70", "80"), ]),
        type = "l", col = "black", xlab = "years", ylab = "rates",
        lwd = 1.5)

## End(Not run)

Compute the link for a given mortality models

Description

Compute the link for a given mortality models

Usage

predictLink(ax, bx, kt, b0x, gc, oxt, ages, years)

Process the initial parameter supplied by the user

Description

Convert the initial parameters supplied by the user into parameters suitable for use within gnm

Usage

processStartValues(object, coefNames, ax, bx, kt, b0x, gc, ages, years, cohorts)

Arguments

Value

a vector of initial parameter estimates

Extract deviance residuals of a Stochastic Mortality Model

Description

Compute deviance residuals of a fitted Stochastic Mortality Model. These residuals can be plotted using plot.resStMoMo.

Usage

## S3 method for class 'fitStMoMo'
residuals(object, scale = TRUE, ...)

Arguments

Value

An object of class "resStMoMo" with the residuals. This object has components:

See Also

plot.resStMoMo

Examples

CBDfit <- fit(cbd(), data = central2initial(EWMaleData), ages.fit = 55:89)
CBDres <- residuals(CBDfit)
plot(CBDres)

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

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)

Do a scatter plot of a matrix according to age-period-cohorts

Description

Do a scatter plot of a matrix according to age-period-cohorts

Usage

scatterplotAPC(mat, ages, years, plotAge = TRUE, plotYear = TRUE,
  plotCohort = TRUE, zeroLine = TRUE, ...)

Arguments

Simulate future sample paths from a Bootstrapped Stochastic Mortality Model

Description

Simulate future sample paths from a Bootstrapped Stochastic Mortality Model. The period indexes $\kappa_t^{(i)}, i = 1,..N,$ are modelled using ether a Multivariate Random Walk with Drift (MRWD) or $N$ independent ARIMA$(p, d, q)$ models. The cohort index $\gamma_{t-x}$ is modelled using an ARIMA$(p, d, q)$. By default an ARIMA$(1, 1, 0)$ with a constant is used.

Usage

## S3 method for class 'bootStMoMo'
simulate(object, nsim = 1, seed = NULL, h = 50,
  oxt = NULL, gc.order = c(1, 1, 0), gc.include.constant = TRUE,
  jumpchoice = c("fit", "actual"), kt.method = c("mrwd", "iarima"),
  kt.order = NULL, kt.include.constant = TRUE, kt.lookback = NULL,
  gc.lookback = NULL, ...)