mev (version 2.1)

xdep.asym: Coefficient of extremal asymmetry

Description

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.

Usage

xdep.asym(
  xdat,
  qlev = NULL,
  nq = 40,
  qlim = c(0.8, 0.99),
  estimator = c("emp", "elik"),
  confint = c("none", "wald", "bootstrap"),
  level = 0.95,
  B = 999L,
  ties.method = "random",
  plot = TRUE,
  ...
)

Value

an invisible data frame with columns

qlev

quantile level of thresholds

coef

extremal asymmetry coefficient estimates

lower

either NULL or a vector containing the lower bound of the confidence interval

upper

either NULL or a vector containing the lower bound of the confidence interval

Arguments

xdat

an n by 2 matrix of observations

qlev

vector of quantile levels at which to evaluate extremal asymmetry

nq

integer; number of quantiles at which to evaluate the coefficient if u is NULL

qlim

a vector of length 2 with the probability limits for the quantiles

estimator

string indicating the estimation method, one of emp or empirical likelihood (elik)

confint

string for the method used to derive confidence intervals, either none (default) or a nonparametric bootstrap

level

probability level for confidence intervals, default to 0.95 or bounds for the interval

B

integer; number of bootstrap replicates (if applicable)

ties.method

string; method for handling ties. See the documentation of rank for available options.

plot

logical; if TRUE, return a plot.

...

additional arguments for backward compatibility

Details

Let U, V be uniform random variables and define the partial extremal dependence coefficients $$\varphi_{+}(u) = \Pr(V > U \mid U > u, V > u),$$, $$\varphi_{-}(u) = \Pr(V < U \mid U > u, V > u),$$ $$\varphi_0(u) = \Pr(V = U \mid U > u, V > u).$$ Define $$ \varphi(u) = \frac{\varphi_{+} - \varphi_{-}}{\varphi_{+} + \varphi_{-}}$$ and the coefficient of extremal asymmetry as \(\varphi = \lim_{u \to 1} \varphi(u)\).

The empirical likelihood estimator, derived for max-stable vectors with unit Frechet margins, is $$\widehat{\varphi}_{\mathrm{el}} = \frac{\sum_i p_i \mathrm{I}(w_i \leq 0.5) - 0.5}{0.5 - 2\sum_i p_i(0.5-w_i) \mathrm{I}(w_i \leq 0.5)}$$ where \(p_i\) is the empirical likelihood weight for observation \(i\), \(\mathrm{I}\) is an indicator function and \(w_i\) is the pseudo-angle associated to the first coordinate, derived based on exceedances above \(u\).

References

Semadeni, C. (2020). Inference on the Angular Distribution of Extremes, PhD thesis, EPFL, no. 8168.

Examples

if (FALSE) {
samp <- rmev(n = 1000,
             d = 2,
             param = 0.2,
             model = "log")
xdep.asym(samp, confint = "wald")
xdep.asym(samp, method = "emplik", confint = "none")
}