mev (version 2.1)

xasym: 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

xasym(
  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

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

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 empirical or empirical likelihood (emplik)

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 parameters for plots

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")
xasym(samp, confint = "wald")
xasym(samp, method = "emplik")
}