Example

We consider Dutch data from CBS; these data were analysed in Einmahl, Einmahl, and Haan (2019). For simplicity, we keep only Dutch people born in the Netherlands, who were at least centenarians when they died and whose death date is known.

thresh0 <- 36525
data(dutch, package = "longevity")
dutch1 <- subset(dutch, ndays > thresh0 & !is.na(ndays) & valid == "A")

We can fit various parametric models accounting for the fact that data are interval truncated. First, we create a list to avoid having to type the name of all arguments repeatedly. These, if not provided directly to function, are selected from the list through arguments.

args <- with(dutch1, list(
  time = ndays,  # time vector
  ltrunc = ltrunc, # left truncation bound
  rtrunc = rtrunc, # right truncation
  thresh = thresh0, # threshold (model only exceedances)
  family = "gp")) # choice of parametric model

The generalized Pareto distribution can be used for extrapolation, provided that the threshold is high enough that shape estimates are more or less stable. To check this, we can produce threshold stability plots, which display point estimates with 95% profile-based pointwise confidence intervals.

tstab_c <- tstab(
  arguments = args,
  family = "gp", # parametric model, here generalized Pareto
  thresh = 102:108 * 365.25, # overwrites thresh
  method = "wald", # type of interval, Wald or profile-likelihood
  plot = FALSE) # by default, calls 'plot' routine
plot(tstab_c, 
     which.plot = "shape", 
     xlab = "threshold (age in days)")

Threshold stability plot with generalized Pareto shape estimates for Dutch data as a function of threshold (in years).

We can fit various parametric models and compare them using the anova call, provided they are nested and share the same data. Diagnostic plots, adapted for survival data, can be used to check goodness-of-fit. These may be computationally intensive to produce in large samples, since they require estimation of the nonparametric maximum likelihood estimator of the distribution function. As such, we pick a relatively high threshold, 108 years, to reduce the computational burden.

(m1 <- fit_elife(arguments = args, 
          thresh = 108 * 365.25,
          family = "gp",
          export = TRUE))

## Model: generalized Pareto distribution. 
## Sampling: interval truncated
## Log-likelihood: -631.242 
## 
## Threshold: 39447 
## Number of exceedances: 90 
## 
## Estimates
##   scale    shape  
## 412.688    0.112  
## 
## Standard Errors
## scale  shape  
## 1.414  0.113  
## 
## Optimization Information
##   Convergence: TRUE

m0 <- fit_elife(arguments = args, 
          thresh = 108 * 365.25,
          family = "exp")
anova(m1, m0)

##     npar Deviance Df     Chisq Pr(>Chisq)
## gp     2 1262.483 NA        NA         NA
## exp    1 1263.112  1 0.6287453  0.4278159

plot(m1, which.plot = "qq")

Quantile-quantile plot of generalized Pareto model to exceedances above 108 years for Dutch.