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"
).
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,
...
)
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
an \(n\) by \(d\) matrix of multivariate observations
vector of percentiles between 0 and 1
number of quantiles of the structural variable at which to form a grid; only used if u = NULL
.
limits for the sequence u
of the structural variable
dependence measure, either of "eta"
or "chi"
named list giving the estimation method for eta
and chi
. Default to "emp"
for both.
string indicating the type of confidence interval for \(\eta\), one of "wald"
or "lrt"
the confidence level required (default to 0.95).
logical indicating whether the estimates and confidence intervals should be truncated in \([0,1]\)
logical indicating whether observations should be transformed to pseudo-uniform scale (default to TRUE
); otherwise, they are assumed to be uniform
string indicating the type of method for rank
; see rank
for a list of options. Default to "random"
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
.
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
chiplot
for bivariate empirical estimates of \(\chi\) and \(\bar{\chi}\).
if (FALSE) {
set.seed(765)
# Max-stable model
dat <- rmev(n = 1000, d = 4, param = 0.7, model = "log")
taildep(dat, confint = 'wald')
}
Run the code above in your browser using DataLab