This function implements estimators of the bivariate coefficient of extremal asymmetry proposed in Semadeni's (2021) PhD thesis. Two estimators are implemented: one based on empirical distributions, the second using empirical likelihood.
xasym(
data,
u = NULL,
nq = 40,
qlim = c(0.8, 0.99),
method = c("empirical", "emplik"),
confint = c("none", "wald", "bootstrap"),
level = 0.95,
B = 999L,
ties.method = "random",
plot = TRUE,
...
)
an invisible data frame with columns
threshold
vector of thresholds on the probability scale
coef
extremal asymmetry coefficient estimates
confint
either NULL
or a matrix with two columns containing the lower and upper bounds for each threshold
an n
by 2 matrix of observations
vector of probability levels at which to evaluate extremal asymmetry
integer; number of quantiles at which to evaluate the coefficient if u
is NULL
a vector of length 2 with the probability limits for the quantiles
string indicating the estimation method, one of empirical
or empirical likelihood (emplik
)
string for the method used to derive confidence intervals, either none
(default) or a nonparametric bootstrap
probability level for confidence intervals, default to 0.95 or bounds for the interval
integer; number of bootstrap replicates (if applicable)
string; method for handling ties. See the documentation of rank for available options.
logical; if TRUE
, return a plot.
additional parameters for plots
Let U
, V
be uniform random variables and define the partial extremal dependence coefficients
\(\varphi_{+}(u) = \Pr(V > U | U > u, V > u)\),
\(\varphi_{-}(u) = \Pr(V < U | U > u, V > u)\) and
\(\varphi_0(u) = \Pr(V = U | U > u, V > u)\)
Define
$$ \varphi(u) = \frac{\varphi_{+} - \varphi_{-}}{\varphi_{+} + \varphi_{-}}$$
The empirical likelihood estimator, derived for max-stable vectors with unit Frechet margins, is $$\frac{\sum_i p_i I(w_i \leq 0.5) - 0.5}{0.5 - 2\sum_i p_i(0.5-w_i) I(w_i \leq 0.5)}$$ where \(p_i\) is the empirical likelihood weight for observation \(i\) and \(w_i\) is the pseudo-angle associated to the first coordinate.
Semadeni, C. (2020). Inference on the Angular Distribution of Extremes, PhD thesis, EPFL, no. 8168.
if (FALSE) {
samp <- rmev(n = 1000,
d = 2,
param = 0.2,
model = "log")
xasym(samp, confint = "wald")
xasym(samp, method = "emplik")
}
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