Empirical Pickands dependence function

Formula: A(t) = -log(P(F_X(X) <= u^(1-t), F_Y(Y) <= u^t)) / (-log u)

Usage

Evpdfn(x, y, t_grid = NULL, u = NULL)

Arguments

x: First variable.
y: Second variable.
t_grid: Points in [0, 1], or NULL for eleven equally spaced.
u: Copula level in (0, 1), or NULL for exp(-1).

Value

List with A, t, estimate (A at 1/2), chi, convex_ok, n, method.

Details

A(t) is read off the empirical copula at the diagonal-shifted point (u^(1-t), u^t). It satisfies A(0) = A(1) = 1 exactly by construction, lies between max(t, 1-t) and 1, equals 1 under independence and max(t, 1-t) under perfect dependence.

References

Pickands (1981), Bull. Int. Statist. Inst. 49:859-878.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Evpdfn(V, V)
#> $A
#>  [1] 1.0000000 0.9808293 0.8000000 0.7000000 0.6931472 0.5000000 0.6931472
#>  [8] 0.7000000 0.8000000 0.9808293 1.0000000
#> 
#> $t
#>  [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
#> 
#> $estimate
#> [1] 0.5
#> 
#> $chi
#> [1] 1
#> 
#> $convex_ok
#> [1] 0
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "empirical Pickands dependence function"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"