Frechet-Hoeffding bounds on a joint distribution

For any joint distribution with the given margins the copula lies between the two bounds, and both are attained: the upper by comonotone variables, the lower by countermonotone ones. Both bounds are themselves copulas in two dimensions, so they satisfy C(u, 1) = u exactly – which is what the tests check, along with the bracketing of the independence copula uv.

Usage

Frdbnd(F_0, F_1, joint = NULL)

Arguments

F_0, F_1

Marginal CDF values in [0, 1], of equal length.

joint

Optional candidate joint CDF at the same points.

Value

List with estimate, lower, upper, independence, width, respects_bounds, n_violations, n, method.

Details

Formula: max(u + v - 1, 0) <= C(u, v) <= min(u, v).

References

Frechet (1951), Sur les tableaux de correlation dont les marges sont donnees, Annales de l'Universite de Lyon A 14:53-77; Hoeffding (1940), Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts der Universitat Berlin 5:181-233.

Examples

Frdbnd(F_0 = c(0.2, 0.5, 0.8), F_1 = c(0.3, 0.6, 0.9))
#> $estimate
#> [1] 0.2
#> 
#> $lower
#> [1] 0.0 0.1 0.7
#> 
#> $upper
#> [1] 0.2 0.5 0.8
#> 
#> $independence
#> [1] 0.06 0.30 0.72
#> 
#> $width
#> [1] 0.2 0.4 0.1
#> 
#> $respects_bounds
#> [1] 1
#> 
#> $n_violations
#> [1] 0
#> 
#> $n
#> [1] 3
#> 
#> $method
#> [1] "max(u + v - 1, 0) <= C(u, v) <= min(u, v), Frechet (1951); Hoeffding (1940)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"