Madogram estimator of the Pickands dependence function

Naveau, Guillou, Cooley and Diebolt (2009), Prop. 3: the lambda-madogram nu(t) = E|F(X)^(1/t) - G(Y)^(1/(1-t))|/2 on rank margins inverts to A(t) = (nu + c)/(1 - nu - c) with c = t/(2(1+t)) + (1-t)/(2(2-t)). A == 1 is asymptotic independence, max(t, 1-t) complete dependence, and every valid A is convex between those envelopes. The estimate is clipped into the envelope as the authors recommend, and the clipping is REPORTED – heavy clipping means the extreme-value model itself fits badly.

Usage

morie_evt_madogram(x, y, t = NULL)

Arguments

x, y: paired observations.
t: evaluation points in (0, 1); a grid when NULL.

Value

list: t, A, A_raw, clipped_fraction, dependence_summary, envelope, n, method.

References

Naveau, Guillou, Cooley and Diebolt (2009), Biometrika 96:1-17, Prop. 3; Pickands (1981).

Examples

set.seed(1)
morie_evt_madogram(stats::rnorm(100), stats::rnorm(100),
  t = c(0.5)
)$A
#> [1] 0.9750064