Monotone functional dependence and the Fréchet bounds #
Nelsen, An Introduction to Copulas, second edition, Theorem 2.5.4 and Corollary 2.5.5
(random-variable version, for laws on Fin 2 → ℝ with continuous marginal CDFs).
For such a law μ, the Sklar copula is the law of (F₀ x₀, F₁ x₁). It equals the comonotonic copula
M exactly when F₀ x₀ = F₁ x₁ almost surely, and the countermonotonic copula W exactly when
F₁ x₁ = 1 - F₀ x₀ almost surely. If the second coordinate is almost surely a strictly increasing
(respectively decreasing) function of the first, these almost-sure identities hold, so the copula
is M (respectively W).
The converse implication is proved as well, which gives the full Theorem 2.5.4: the copula is
M if and only if x₁ = f x₀ almost surely for a function f that is monotone on a set carrying
the law of x₀ (ofContinuousMarginals_eq_comonotonic_iff_exists_monotoneOn), and W if and
only if the same holds with an antitone f
(ofContinuousMarginals_eq_countermonotonic_iff_exists_antitoneOn). The function is explicit:
f = G₁ ∘ F₀ (respectively f = G₁ ∘ (1 - F₀)), with G₁ = realQuantile the quantile of the
second marginal, and the carrier set is {x | 0 < F₀ x < 1}.
Monotonicity cannot in general be required on all of ℝ: if x₀ is uniform on [0,1] and
x₁ = Φ⁻¹(x₀) is standard normal, then the copula is M, but a monotone f : ℝ → ℝ with
x₁ = f x₀ almost surely would have to be -∞ on (-∞, 0). Nor can strict monotonicity be
required, since F₀ is constant on gaps in the support of x₀. For globally monotone f the
sufficiency direction is eq_comonotonic_of_ae_eq_monotone.
The Sklar copula of a bivariate law with continuous marginals is M if and only if the
probability integral transforms of the two coordinates agree almost surely.
The Sklar copula of a bivariate law with continuous marginals is W if and only if the
second probability integral transform is almost surely one minus the first.
If the second coordinate is almost surely g of the first, its law is the image law.
Nelsen, Theorem 2.5.4 (sufficiency, increasing case). If the second coordinate is almost
surely a strictly increasing function of the first, the Sklar copula is M.
Nelsen, Theorem 2.5.4 (sufficiency, decreasing case). If the second coordinate is almost
surely a strictly decreasing function of the first, the Sklar copula is W.
Any Sklar copula of a law with continuous marginals whose second coordinate is almost surely
a strictly increasing function of the first is M.
Any Sklar copula of a law with continuous marginals whose second coordinate is almost surely
a strictly decreasing function of the first is W.
Monotone functions on a carrier set: the full Theorem 2.5.4 #
A coordinate of a law with continuous marginal CDF takes each fixed value with probability zero.
If x₁ = f x₀ almost surely with f monotone on a set carrying x₀, the marginal CDFs are
linked by F₁ (f y) = F₀ y on that set.
If x₁ = f x₀ almost surely with f antitone on a set carrying x₀, the marginal CDFs are
linked by F₁ (f y) = 1 - F₀ y on that set.
Nelsen, Theorem 2.5.4 (sufficiency, increasing case, general form). If the second
coordinate is almost surely f of the first, with f nondecreasing on a set carrying the first
coordinate, the Sklar copula is M.
Nelsen, Theorem 2.5.4 (sufficiency, decreasing case, general form). If the second
coordinate is almost surely f of the first, with f nonincreasing on a set carrying the first
coordinate, the Sklar copula is W.
Sufficiency with a globally nondecreasing function.
Sufficiency with a globally nonincreasing function.
Almost surely the first probability integral transform lies strictly inside (0, 1).
Almost surely the quantile of the second marginal inverts its CDF.
Nelsen, Theorem 2.5.4 (necessity, increasing case). If the Sklar copula is M, then
x₁ = G₁ (F₀ x₀) almost surely, where G₁ is the quantile of the second marginal.
Nelsen, Theorem 2.5.4 (necessity, decreasing case). If the Sklar copula is W, then
x₁ = G₁ (1 - F₀ x₀) almost surely, where G₁ is the quantile of the second marginal.
Nelsen, Theorem 2.5.4 (increasing case). For a law with continuous marginals, the Sklar
copula is M if and only if the second coordinate is almost surely a nondecreasing function of
the first, where the function need only be nondecreasing on a set carrying the first
coordinate.
Nelsen, Theorem 2.5.4 (decreasing case). For a law with continuous marginals, the Sklar
copula is W if and only if the second coordinate is almost surely a nonincreasing function of
the first, where the function need only be nonincreasing on a set carrying the first
coordinate.
Theorem 2.5.4 for an arbitrary Sklar copula of a law with continuous marginals (increasing case).
Theorem 2.5.4 for an arbitrary Sklar copula of a law with continuous marginals (decreasing case).