Coefficient of tail correlation and tail dependence

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 confint obtained through a likelihood ratio test statistic ("lrt") or the Wald statistic ("wald").

Usage

taildep(
  data,
  u = NULL,
  nq = 40,
  qlim = c(0.8, 0.99),
  depmeas = c("eta", "chi"),
  method = list(eta = c("emp", "betacop", "gpd", "hill"), chi = c("emp", "betacop")),
  confint = c("wald", "lrt"),
  level = 0.95,
  trunc = TRUE,
  empirical.transformation = TRUE,
  ties.method = "random",
  plot = TRUE,
  ...
)

Arguments

data: an $n$ by $d$ matrix of multivariate observations
u: 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"
method: 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 $[0,1]$
empirical.transformation: logical indicating whether observations should be transformed to pseudo-uniform scale (default to TRUE); otherwise, they are assumed to be uniform
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.

Value

a named list with elements

u: 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

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 confint 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

Note

As of version 1.15, the percentiles used are from the minimum variable. This ensures that, regardless of the number of variables, there is no error message returned because the quantile levels are too low for there to be observations

Examples

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