taildep: Coefficient of tail correlation and tail dependence

Description

For data with unit Pareto margins, the coefficient of tail dependence $\eta$ is defined via $$\Pr(\min(X) > x) = L(x)x^{-1/\eta},$$ where $L(x)$ is a slowly varying function. Ignoring the latter, several estimators of $\eta$ can be defined. In unit Pareto margins, $\eta$ is a nonnegative shape parameter that can be estimated by fitting a generalized Pareto distribution above a high threshold. In exponential margins, $\eta$ is a scale parameter and the maximum likelihood estimator of the latter is the Hill estimator. Both methods are based on peaks-over-threshold and the user can choose between pointwise confidence obtained through a likelihood ratio test statistic ("lrt") or the Wald statistic ("wald").

Usage

taildep(
  xdat,
  qlev = NULL,
  nq = 40,
  qlim = c(0.8, 0.99),
  depmeas = c("eta", "chi"),
  estimator = list(eta = c("emp", "betacop", "gpd", "hill"), chi = c("emp", "betacop")),
  confint = c("wald", "lrt"),
  level = 0.95,
  trunc = TRUE,
  margtrans = c("emp", "none"),
  ties.method = "random",
  plot = TRUE,
  ...
)

Value

a named list with elements

qlev: a K vector of percentile levels
eta: a K by 3 matrix with point estimates, lower and upper confidence intervals
chi: a K by 3 matrix with point estimates, lower and upper confidence intervals

Arguments

xdat: an $n$ by $d$ matrix of multivariate observations
qlev: vector of percentiles between 0 and 1
nq: number of quantiles of the structural variable at which to form a grid; only used if u = NULL.
qlim: limits for the sequence u of the structural variable
depmeas: dependence measure, either of "eta" or "chi"
estimator: named list giving the estimation method for eta and chi. Default to "emp" for both.
confint: string indicating the type of confidence interval for $\eta$, one of "wald" or "lrt"
level: the confidence level required (default to 0.95).
trunc: logical indicating whether the estimates and confidence intervals should be truncated in $[0,1]$
margtrans: marginal transformation; if "none", data are assumed to be in uniform margins
ties.method: string indicating the type of method for rank; see rank for a list of options. Default to "random"
plot: logical; should graphs be plotted?
...: additional arguments passed to plot; current support for main, xlab, ylab, add and further pch, lty, type, col for points; additional arguments for confidence intervals are handled via cipch, cilty, citype, cicol.

Details

The most common approach for estimation is the empirical survival copula, by evaluating the proportion of sample minima with uniform margins that exceed a given $x$. An alternative estimator uses a smoothed estimator of the survival copula using Bernstein polynomial, resulting in the so-called betacop estimator. Approximate pointwise confidence intervals for the latter are obtained by assuming the proportion of points is binomial.

The coefficient of tail correlation $\chi$ is $$\chi = \lim_{u \to 1} \frac{\Pr(F_1(X_1)>u, \ldots, F_D(X_D)>u)}{1-u}.$$ Asymptotically independent vectors have $\chi = 0$. The estimator uses an estimator of the survival copula

Examples

Run this code

if (FALSE) {
set.seed(765)
# Max-stable model
dat <- rmev(n = 1000, d = 4, param = 0.7, model = "log")
taildep(dat, confint = 'wald')
}