mev (version 2.1)

xdep.eta: Coefficient of 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

xdep.eta(
  xdat,
  qlev = NULL,
  nq = 40,
  qlim = c(0.8, 0.99),
  estimator = c("emp", "betacop", "gpd", "hill", "kj"),
  confint = c("wald", "lrt"),
  level = 0.95,
  margtrans = c("emp", "sp", "none"),
  ties.method = "random",
  plot = TRUE,
  mqlev = NULL,
  ...
)

Value

a data frame

• qlev: quantile level of estimates

• coef: point estimates

• lower: lower bound of confidence interval

• upper: lower bound of confidence interval

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

estimator

string giving estimator to employ

confint

string indicating the type of confidence interval, one of "wald" or "lrt"

level

the confidence level required (default to 0.95).

margtrans

string giving the marginal transformation, one of emp for rank-based transformation or none if data are already on the uniform scale

ties.method

string indicating the type of method for rank; see rank for a list of options. Default to "random"

plot

logical; if TRUE, return a plot

mqlev

marginal quantile levels for semiparametric estimation for estimator kj; data above this are modelled using a generalized Pareto distribution. If missing, empirical estimation is used throughout

...

additional arguments to taildep, currently ignored

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.

References

Ledford, A.W. and J. A. Tawn (1996), Statistics for near independence in multivariate extreme values. Biometrika, 83(1), 169--187.

Examples

if (FALSE) {
set.seed(765)
# Max-stable model
dat <- rmev(n = 1000, d = 2, param = 0.7, model = "log")
xdep.eta(dat, confint = 'wald')
}