Coefficient of extremal asymmetry

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(
  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,
  ...
)

Arguments

data: an n by 2 matrix of observations
u: vector of probability 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
method: 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

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

Details

Let U, V be uniform random variables and define the partial extremal dependence coefficients $$\varphi_{+}(u) = \Pr(V > U | U > u, V > u),$$, $$\varphi_{-}(u) = \Pr(V < U | U > u, V > u),$$ $$\varphi_0(u) = \Pr(V = U | U > u, V > u).$$ 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 $$\frac{\sum_i p_i I(w_i \leq 0.5) - 0.5}{0.5 - 2\sum_i p_i(0.5-w_i) I(w_i \leq 0.5)}$$ where $p_i$ is the empirical likelihood weight for observation $i$ and $w_i$ is the pseudo-angle associated to the first coordinate.

References

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

Examples

if (FALSE) { # \dontrun{
samp <- rmev(n = 1000,
             d = 2,
             param = 0.2,
             model = "log")
xasym(samp, confint = "wald")
xasym(samp, method = "emplik")
} # }