chi: Tail Dependence Coefficient (Chi Statistic)

In eFCM: Exponential Factor Copula Model

Tail Dependence Coefficient (Chi Statistic)

Description

Compute the conditional exceedance probability \chi_h(u), either from a fitted eFCM model (chi.fcm) or empirically (Echi). \chi_h(u) measures the probability of simultaneous exceedances at high but finite thresholds.

Usage

## S3 method for class 'fcm'
chi(object, h, u = 0.95, ...)

Echi(object, which = c(1, 2), u = 0.95)

Arguments

object	an object of class "fcm", created by fcm().
h	a positive numeric value representing the spatial distance (in kilometers).
u	a numeric value between 0 and 1 specifying the quantile threshold. Default is 0.95.
...	currently ignored.
which	A length-two integer vector giving the indices of the columns in object$data to be used for the empirical chi calculation.

Details

For two locations s_1 and s_2 separated by distance h, with respective vector components W(s_1) and W(s_2), the conditional exceedance probability is defined as

\chi_h(u) \;=\; \lim_{u \to 1} \Pr\!\big( F_{s_1}(W(s_1)) > u \;\mid\; F_{s_2}(W(s_2)) > u \big).

For the eFCM, the conditional exceedance probability \chi_{\mathrm{eFCM}}(u) can be computed as

\chi_{\mathrm{eFCM}}(u) = \frac{1 - 2u + \Phi_2\big(z(u), z(u); \rho\big) - 2 \exp\!\left( \frac{\lambda^2}{2} - \lambda\, z(u)\, \Phi_2\big(q; 0, \Omega\big) \right)} {1 - u}.

Here, z(u) = F_1^{-1}(u; \lambda) is the marginal quantile function, \Phi_2(\cdot,\cdot;\rho) denotes the bivariate standard normal CDF with correlation \rho, q = \lambda(1-\rho), and \Omega is the correlation matrix.

Value

A named numeric value, the chi statistic for given h and u.

Methods

• chi.fcm(): Model-based estimate from an object of class "fcm".

• Echi(): Empirical estimate.

References

Castro-Camilo, D. and Huser, R. (2020). Local likelihood estimation of complex tail dependence structures, applied to US precipitation extremes. Journal of the American Statistical Association, 115(531), 1037–1054.

Examples

fit <- fcm(...)
chi(fit, h = 150, u = 0.95)

eFCM documentation built on Sept. 9, 2025, 5:52 p.m.