game: Smooth Parameter Estimation and Bootstrapping of Generalized Pareto Distributions with Penalized Maximum Likelihood Estimation

Description

gamGPDfit() fits the parameters of a generalized Pareto distribution (GPD) depending on covariates in a non- or semiparametric way.

gamGPDboot() fits and bootstraps the parameters of a GPD distribution depending on covariates in a non- or semiparametric way. Applies the post-blackend bootstrap of Chavez-Demoulin and Davison (2005).

Usage

gamGPDfit(x, threshold, nextremes = NULL, datvar, xiFrhs, nuFrhs,
          init = fit.GPD(x[,datvar], threshold = threshold,
                         type = "pwm", verbose = FALSE)$par.ests,
          niter = 32, include.updates = FALSE, eps.xi = 1e-05, eps.nu = 1e-05,
          progress = TRUE, adjust = TRUE, verbose = FALSE, …)
gamGPDboot(x, B, threshold, nextremes = NULL, datvar, xiFrhs, nuFrhs,
           init = fit.GPD(x[,datvar], threshold = threshold,
                          type = "pwm", verbose = FALSE)$par.ests,
           niter = 32, include.updates = FALSE, eps.xi = 1e-5, eps.nu = 1e-5,
           boot.progress = TRUE, progress = FALSE, adjust = TRUE, verbose = FALSE,
           debug = FALSE, …)

Arguments

data.frame containing the losses (in some component; can be specified with the argument datvar; the other components contain the covariates).

number of bootstrap replications.

threshold

threshold of the peaks-over-threshold (POT) method.

nextremes

number of excesses. This can be used to determine

datvar

name of the data column in x which contains the the data to be modeled.

xiFrhs

right-hand side of the formula for $\xi$ in the gam() call for fitting $\xi$.

nuFrhs

right-hand side of the formula for $\nu$ in the gam() call for fitting $\nu$.

init

bivariate vector containing initial values for $(\xi, \beta)$.

niter

maximal number of iterations in the backfitting algorithm.

include.updates

logical indicating whether updates for xi and nu are returned as well (note: this might lead to objects of large size).

eps.xi

epsilon for stop criterion for $\xi$.

eps.nu

epsilon for stop criterion for $\nu$.

boot.progress

logical indicating whether progress information about gamGPDboot() is displayed.

progress

logical indicating whether progress information about gamGPDfit() is displayed. For gamGPDboot(), progress is only passed to gamGPDfit() in the case that boot.progress==TRUE.

adjust

logical indicating whether non-real values of the derivatives are adjusted.

verbose

logical indicating whether additional information (in case of undesired behavior) is printed. For gamGPDboot(), progress is only passed to gamGPDfit() if boot.progress==TRUE.

debug

logical indicating whether initial fit (before the bootstrap is initiated) is saved.

…

additional arguments passed to gam() (which is called internally; see the source code of gamGPDfitUp()).

Value

gamGPDfit() returns either an empty list (list(); in case at least one of the two gam() calls in the internal function gamGPDfitUp() fails) or a list with the components

xi:: estimated parameters $\xi$;
beta:: estimated parameters $\beta$;
nu:: estimated parameters $\nu$;
se.xi:: standard error for $\xi$ ((possibly adjusted) second-order derivative of the reparameterized log-likelihood with respect to $\xi$) multiplied by -1;
se.nu:: standard error for $\nu$ ((possibly adjusted) second-order derivative of the reparameterized log-likelihood with respect to $\nu$) multiplied by -1;
xi.covar:: (unique) covariates for $\xi$;
nu.covar:: (unique) covariates for $\nu$;
covar:: available covariate combinations used for fitting $\beta$ ($\xi$, $\nu$);
y:: vector of excesses (exceedances minus threshold);
res:: residuals;
MRD:: mean relative distances between for all iterations, calculated between old parameters $(\xi, \nu)$ (from the last iteration) and new parameters (currently estimated ones);
logL:: log-likelihood at the estimated parameters;
xiObj:: R object of type gamObject for estimated $\xi$ (returned by mgcv::gam());
nuObj:: R object of type gamObject for estimated $\nu$ (returned by mgcv::gam());
xiUpdates:: if include.updates is TRUE, updates for $\xi$ for each iteration. This is a list of R objects of type gamObject which contains xiObj as last element;
nuUpdates:: if include.updates is TRUE, updates for $\nu$ for each iteration. This is a list of R objects of type gamObject which contains nuObj as last element;

gamGPDboot() returns a list of length B+1 where the first component contains the results of the initial fit via gamGPDfit() and the other B components contain the results for each replication of the post-blackend bootstrap. Components for which gam() fails (e.g., due to too few data) are given as empty lists (list()).

Details

gamGPDfit() fits the parameters $\xi$ and $\beta$ of the generalized Pareto distribution $\mathrm{GPD}(\xi,\beta)$ depending on covariates in a non- or semiparametric way. The distribution function is given by $$G_{\xi,\beta}(x)=1-(1+\xi x/\beta)^{-1/\xi},\quad x\ge0,$$ for $\xi>0$ (which is what we assume) and $\beta>0$. Note that $\beta$ is also denoted by $\sigma$ in this package. Estimation of $\xi$ and $\beta$ by gamGPDfit() is done via penalized maximum likelihood estimation, where the estimators are computed with a backfitting algorithm. In order to guarantee convergence of this algorithm, a reparameterization of $\beta$ in terms of the parameter $\nu$ is done via $$\beta=\exp(\nu)/(1+\xi).$$ The parameters $\xi$ and $\nu$ (and thus $\beta$) are allowed to depend on covariates (including time) in a non- or semiparametric way, for example: $$\xi=\xi(\bm{x},t)=\bm{x}^{\top}\bm{\alpha}_{\xi}+h_{\xi}(t),$$ $$\nu=\nu(\bm{x},t)=\bm{x}^{\top}\bm{\alpha}_{\nu}+h_{\nu}(t),$$ where $\bm{x}$ denotes the vector of covariates, $\bm{\alpha}_{\xi}$, $\bm{\alpha}_{\nu}$ are parameter vectors and $h_{\xi}$, $h_{\nu}$ are regression splines. For more details, see the references and the source code.

gamGPDboot() first fits the GPD parameters via gamGPDfit(). It then conducts the post-blackend bootstrap of Chavez-Demoulin and Davison (2005). To this end, it computes the residuals, resamples them (B times), reconstructs the corresponding excesses, and refits the GPD parameters via gamGPDfit() again.

Note that if gam() fails in gamGPDfit() or the fitting or one of the bootstrap replications in gamGPDboot(), then the output object contains (an) empty (sub)list(s). These failures typically happen for too small sample sizes.

References

Chavez-Demoulin, V., and Davison, A. C. (2005). Generalized additive models for sample extremes. Applied Statistics 54(1), 207--222.

Chavez-Demoulin, V., Embrechts, P., and Hofert, M. (2014). An extreme value approach for modeling Operational Risk losses depending on covariates. Journal of Risk and Insurance; accepted.

Examples

Run this code

# NOT RUN {
library(QRM)
## generate an example data set
years <- 2003:2012 # years
nyears <- length(years)
n <- 250 # sample size for each (different) xi
u <- 200 # threshold
rGPD <- function(n, xi, beta) ((1-runif(n))^(-xi)-1)*beta/xi # sampling GPD

set.seed(17) # setting seed
xi.true.A <- seq(0.4, 0.8, length=nyears) # true xi for group "A"
## generate losses for group "A"
lossA <- unlist(lapply(1:nyears,
                       function(y) u + rGPD(n, xi=xi.true.A[y], beta=1)))
xi.true.B <- xi.true.A^2 # true xi for group "B"
## generate losses for group "B"
lossB <- unlist(lapply(1:nyears,
                       function(y) u + rGPD(n, xi=xi.true.B[y], beta=1)))
## build data frame
time <- rep(rep(years, each=n), 2) # "2" stands for the two groups
covar <- rep(c("A","B"), each=n*nyears)
value <- c(lossA, lossB)
x <- data.frame(covar=covar, time=time, value=value)

## fit
eps <- 1e-3 # to decrease the run time for this example
require(mgcv) # due to s()
fit <- gamGPDfit(x, threshold=u, datvar="value", xiFrhs=~covar+s(time)-1,
                 nuFrhs=~covar+s(time)-1, eps.xi=eps, eps.nu=eps)
## note: choosing s(..., bs="cr") will fit cubic splines

## grab the fitted values per group and year
xi.fit <- fitted(fit$xiObj)
xi.fit. <- xi.fit[1+(0:(2*nyears-1))*n] # pick fit for each group and year
xi.fit.A <- xi.fit.[1:nyears] # fit for "A" and each year
xi.fit.B <- xi.fit.[(nyears+1):(2*nyears)] # fit for "B" and each year

## plot the fitted values of xi and the true ones we simulated from
par(mfrow=c(1,2))
plot(years, xi.true.A, type="l", ylim=range(xi.true.A, xi.fit.A),
     main="Group A", xlab="Year", ylab=expression(xi))
points(years, xi.fit.A, type="l", col="red")
legend("topleft", inset=0.04, lty=1, col=c("black", "red"),
       legend=c("true", "fitted"), bty="n")
plot(years, xi.true.B, type="l", ylim=range(xi.true.B, xi.fit.B),
     main="Group B", xlab="Year", ylab=expression(xi))
points(years, xi.fit.B, type="l", col="blue")
legend("topleft", inset=0.04, lty=1, col=c("black", "blue"),
       legend=c("true", "fitted"), bty="n")
# }

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